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Mathematics LibreTexts

7-1. Solving Trigonometric Equations with Identities

Solving Trigonometric Equations with Identities
In this section, you will:
  • Verify the fundamental trigonometric identities.
  • Simplify trigonometric expressions using algebra and the identities.
<figure class="small" id="Figure_07_01_006" style="color: rgb(0, 0, 0); font-family: 'Times New Roman'; font-size: medium; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: auto; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; widows: 1; word-spacing: 0px; -webkit-text-stroke-width: 0px;"> <figcaption>International passports and travel documents</figcaption> Photo of international passports.</figure>

In espionage movies, we see international spies with multiple passports, each claiming a different identity. However, we know that each of those passports represents the same person. The trigonometric identities act in a similar manner to multiple passports—there are many ways to represent the same trigonometric expression. Just as a spy will choose an Italian passport when traveling to Italy, we choose the identity that applies to the given scenario when solving a trigonometric equation.

In this section, we will begin an examination of the fundamental trigonometric identities, including how we can verify them and how we can use them to simplify trigonometric expressions.

Verifying the Fundamental Trigonometric Identities

Identities enable us to simplify complicated expressions. They are the basic tools of trigonometry used in solving trigonometric equations, just as factoring, finding common denominators, and using special formulas are the basic tools of solving algebraic equations. In fact, we use algebraic techniques constantly to simplify trigonometric expressions. Basic properties and formulas of algebra, such as the difference of squares formula and the perfect squares formula, will simplify the work involved with trigonometric expressions and equations. We already know that all of the trigonometric functions are related because they all are defined in terms of the unit circle. Consequently, any trigonometric identity can be written in many ways.

To verify the trigonometric identities, we usually start with the more complicated side of the equation and essentially rewrite the expression until it has been transformed into the same expression as the other side of the equation. Sometimes we have to factor expressions, expand expressions, find common denominators, or use other algebraic strategies to obtain the desired result. In this first section, we will work with the fundamental identities: the Pythagorean identities, the even-odd identities, the reciprocal identities, and the quotient identities.

We will begin with the Pythagorean identities (see [link]), which are equations involving trigonometric functions based on the properties of a right triangle. We have already seen and used the first of these identifies, but now we will also use additional identities.

 
Pythagorean Identities
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msup><mrow><mi>sin</mi></mrow></msup></mrow></annotation-xml></semantics></math> 2 θ+ cos 2 θ=1 <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>1</mn><mo>+</mo><msup/></mrow></annotation-xml></semantics></math> cot 2 θ= csc 2 θ <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>1</mn><mo>+</mo><msup/></mrow></annotation-xml></semantics></math> tan 2 θ= sec 2 θ

The second and third identities can be obtained by manipulating the first. The identity<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>1</mn><mo>+</mo><msup/></mrow></annotation-xml></semantics></math> cot 2 θ= csc 2 θ is found by rewriting the left side of the equation in terms of sine and cosine.

Prove:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>1</mn><mo>+</mo><msup/></mrow></annotation-xml></semantics></math> cot 2 θ= csc 2 θ

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mn>1</mn><mo>+</mo><msup/></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> cot 2 θ=( 1+ cos 2 θ sin 2 θ ) Rewrite the left side.                  =( sin 2 θ sin 2 θ )+( cos 2 θ sin 2 θ )Write both terms with the common denominator.                  = sin 2 θ+ cos 2 θ sin 2 θ                  = 1 sin 2 θ                  = csc 2θ

Similarly,<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>1</mn><mo>+</mo><msup/></mrow></annotation-xml></semantics></math> tan 2 θ= sec 2 θ can be obtained by rewriting the left side of this identity in terms of sine and cosine. This gives

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mn>1</mn><mo>+</mo><msup/></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> tan 2 θ=1+ ( sin θ cos θ ) 2 Rewrite left side.                        = ( cos θ cos θ ) 2 + ( sin θ cos θ ) 2Write both terms with the common denominator.                        = cos 2  θ+ sin 2  θ cos 2  θ                        = 1 cos 2  θ                        =sec 2  θ

The next set of fundamental identities is the set of even-odd identities. The even-odd identities relate the value of a trigonometric function at a given angle to the value of the function at the opposite angle and determine whether the identity is odd or even. (See [link]).

Even-Odd Identities
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mi>tan</mi><mo stretchy="false">(</mo><mo>−</mo><mi>θ</mi><mo stretchy="false">)</mo><mo>=</mo><mo>−</mo><mi>tan</mi><mtext> </mtext><mi>θ</mi></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> cot(−θ)=−cot θ <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mi>sin</mi><mo stretchy="false">(</mo><mo>−</mo><mi>θ</mi><mo stretchy="false">)</mo><mo>=</mo><mo>−</mo><mi>sin</mi><mtext> </mtext><mi>θ</mi></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> csc(−θ)=−csc θ <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mi>cos</mi><mo stretchy="false">(</mo><mo>−</mo><mi>θ</mi><mo stretchy="false">)</mo><mo>=</mo><mi>cos</mi><mtext> </mtext><mi>θ</mi></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> sec(−θ)=sec θ

Recall that an odd function is one in which<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>f</mi><mrow><mo>(−</mo></mrow></mrow></annotation-xml></semantics></math> x )= −f( x ) for all<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>x</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>in the domain of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>f</mi><mo>.</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>The sine function is an odd function because<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>sin</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> −θ )=−sin θ. The graph of an odd function is symmetric about the origin. For example, consider corresponding inputs of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mfrac/></mrow></annotation-xml></semantics></math> π 2  and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mo>−</mo><mfrac/></mrow></annotation-xml></semantics></math> π 2 . The output of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>sin</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> π 2 ) is opposite the output of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>sin</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> − π 2 ). Thus,

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mi>sin</mi><mrow><mo>(</mo></mrow></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> π 2 )=1                         and    sin( − π 2 )=−sin( π 2 )                       =−1

This is shown in [link].

<figure class="small" id="Figure_07_01_002"> <figcaption>Graph of <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>y</mi><mo>=</mo><mi>sin</mi><mtext> </mtext><mi>θ</mi></mrow></annotation-xml></semantics></math></figcaption> Graph of y=sin(theta) from -2pi to 2pi, showing in particular that it is symmetric about the origin. Points given are (pi/2, 1) and (-pi/2, -1).</figure>

Recall that an even function is one in which

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> −x )=f( x ) for all x in the domain of f

The graph of an even function is symmetric about the y-axis. The cosine function is an even function because<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>cos</mi><mo stretchy="false">(</mo><mo>−</mo><mi>θ</mi><mo stretchy="false">)</mo><mo>=</mo><mi>cos</mi><mtext> </mtext><mi>θ</mi><mo>.</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>For example, consider corresponding inputs<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mfrac/></mrow></annotation-xml></semantics></math> π 4  and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mo>−</mo><mfrac/></mrow></annotation-xml></semantics></math> π 4 . The output of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>cos</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> π 4 ) is the same as the output of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>cos</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> − π 4 ). Thus,

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mi>cos</mi><mrow><mo>(</mo></mrow></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> − π 4 )=cos( π 4 )               ≈0.707

See [link].

<figure class="small" id="Figure_07_01_003"> <figcaption>Graph of <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>y</mi><mo>=</mo><mi>cos</mi><mtext> </mtext><mi>θ</mi></mrow></annotation-xml></semantics></math></figcaption> Graph of y=cos(theta) from -2pi to 2pi, showing in particular that it is symmetric about the y-axis. Points given are (-pi/4, .707) and (pi/4, .707). </figure>

For all<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>θ</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>in the domain of the sine and cosine functions, respectively, we can state the following:

  • Since<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>sin</mi><mrow><mo>(−</mo><mrow><mi>θ</mi></mrow></mrow></mrow></annotation-xml></semantics></math> )=−sin θ, sine is an odd function.
  • Since,<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>cos</mi><mrow><mo>(−</mo></mrow></mrow></annotation-xml></semantics></math> θ )=cos θ, cosine is an even function.

The other even-odd identities follow from the even and odd nature of the sine and cosine functions. For example, consider the tangent identity,<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>tan</mi><mrow><mo>(−</mo></mrow></mrow></annotation-xml></semantics></math> θ )=−tan θ. We can interpret the tangent of a negative angle as<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>tan</mi><mrow><mo>(−</mo></mrow></mrow></annotation-xml></semantics></math> θ )= sin( −θ ) cos(− θ ) = −sin θ cos θ =−tan θ. Tangent is therefore an odd function, which means that<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>tan</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> −θ )=−tan( θ ) for all<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>θ</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>in the domain of the tangent function.

The cotangent identity,<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>cot</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> −θ )=−cot θ, also follows from the sine and cosine identities. We can interpret the cotangent of a negative angle as<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>cot</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> −θ )= cos( −θ ) sin( −θ ) = cos θ −sin θ =−cot θ. Cotangent is therefore an odd function, which means that<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>cot</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> −θ )=−cot( θ ) for all<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>θ</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>in the domain of the cotangent function.

The cosecant function is the reciprocal of the sine function, which means that the cosecant of a negative angle will be interpreted as<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>csc</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> −θ )= 1 sin( −θ ) = 1 −sin θ =−csc θ. The cosecant function is therefore odd.

Finally, the secant function is the reciprocal of the cosine function, and the secant of a negative angle is interpreted as<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>sec</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> −θ )= 1 cos( −θ ) = 1 cos θ =sec θ. The secant function is therefore even.

To sum up, only two of the trigonometric functions, cosine and secant, are even. The other four functions are odd, verifying the even-odd identities.

The next set of fundamental identities is the set of reciprocal identities, which, as their name implies, relate trigonometric functions that are reciprocals of each other. See [link].

Reciprocal Identities
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>sin</mi><mtext> </mtext><mi>θ</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 1 csc θ <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>csc</mi><mtext> </mtext><mi>θ</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 1 sin θ
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>cos</mi><mtext> </mtext><mi>θ</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 1 sec θ <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>sec</mi><mtext> </mtext><mi>θ</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 1 cos θ
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>tan</mi><mtext> </mtext><mi>θ</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 1 cot θ <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>cot</mi><mtext> </mtext><mi>θ</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 1 tan θ

The final set of identities is the set of quotient identities, which define relationships among certain trigonometric functions and can be very helpful in verifying other identities. See [link].

 
Quotient Identities
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>tan</mi><mtext> </mtext><mi>θ</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> sin θ cos θ <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>cot</mi><mtext> </mtext><mi>θ</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> cos θ sin θ

The reciprocal and quotient identities are derived from the definitions of the basic trigonometric functions.

Summarizing Trigonometric Identities

The Pythagorean identities are based on the properties of a right triangle.

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msup><mrow><mi>cos</mi></mrow></msup></mrow></annotation-xml></semantics></math> 2 θ+ sin 2 θ=1
<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>1</mn><mo>+</mo><msup/></mrow></annotation-xml></semantics></math> cot 2 θ= csc 2 θ
<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>1</mn><mo>+</mo><msup/></mrow></annotation-xml></semantics></math> tan 2 θ= sec 2 θ

The even-odd identities relate the value of a trigonometric function at a given angle to the value of the function at the opposite angle.

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>tan</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> −θ )=−tan θ
<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>cot</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> −θ )=−cot θ
<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>sin</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> −θ )=−sin θ
<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>csc</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> −θ )=−csc θ
<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>cos</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> −θ )=cos θ
<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>sec</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> −θ )=sec θ

The reciprocal identities define reciprocals of the trigonometric functions.

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>sin</mi><mtext> </mtext><mi>θ</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 1 csc θ
<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>cos</mi><mtext> </mtext><mi>θ</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 1 sec θ
<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>tan</mi><mtext> </mtext><mi>θ</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 1 cot θ
<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>csc</mi><mtext> </mtext><mi>θ</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 1 sin θ
<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>sec</mi><mtext> </mtext><mi>θ</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 1 cos θ
<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>cot</mi><mtext> </mtext><mi>θ</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 1 tan θ

The quotient identities define the relationship among the trigonometric functions.

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>tan</mi><mtext> </mtext><mi>θ</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> sin θ cos θ
<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>cot</mi><mtext> </mtext><mi>θ</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> cos θ sin θ
Graphing the Equations of an Identity

Graph both sides of the identity<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>cot</mi><mtext> </mtext><mi>θ</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 1 tan θ . In other words, on the graphing calculator, graph<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>y</mi><mo>=</mo><mi>cot</mi><mtext> </mtext><mi>θ</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>y</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 1 tan θ .

See [link].

<figure class="small" id="Figure_07_01_007">Graph of y = cot(theta) and y=1/tan(theta) from -2pi to 2pi. They are the same!</figure>
Analysis

We see only one graph because both expressions generate the same image. One is on top of the other. This is a good way to prove any identity. If both expressions give the same graph, then they must be identities.

Given a trigonometric identity, verify that it is true.

  1. Work on one side of the equation. It is usually better to start with the more complex side, as it is easier to simplify than to build.
  2. Look for opportunities to factor expressions, square a binomial, or add fractions.
  3. Noting which functions are in the final expression, look for opportunities to use the identities and make the proper substitutions.
  4. If these steps do not yield the desired result, try converting all terms to sines and cosines.
Verifying a Trigonometric Identity

Verify<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>tan</mi><mtext> </mtext><mi>θ</mi><mi>cos</mi><mtext> </mtext><mi>θ</mi><mo>=</mo><mi>sin</mi><mtext> </mtext><mi>θ</mi><mo>.</mo></mrow></annotation-xml></semantics></math>

We will start on the left side, as it is the more complicated side:

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mi>tan</mi><mtext> </mtext><mi>θ</mi><mi>cos</mi><mtext> </mtext><mi>θ</mi><mo>=</mo><mrow><mo>(</mo></mrow></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> sin θ cos θ )cos θ                 =( sin θ cos θ ) cos θ                 =sin θ
Analysis

This identity was fairly simple to verify, as it only required writing<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>tan</mi><mtext> </mtext><mi>θ</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>in terms of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>sin</mi><mtext> </mtext><mi>θ</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>cos</mi><mtext> </mtext><mi>θ</mi><mo>.</mo></mrow></annotation-xml></semantics></math>

Verify the identity<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>csc</mi><mtext> </mtext><mi>θ</mi><mi>cos</mi><mtext> </mtext><mi>θ</mi><mi>tan</mi><mtext> </mtext><mi>θ</mi><mo>=</mo><mn>1.</mn></mrow></annotation-xml></semantics></math>

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mi>csc</mi><mtext> </mtext><mi>θ</mi><mi>cos</mi><mtext> </mtext><mi>θ</mi><mi>tan</mi><mtext> </mtext><mi>θ</mi><mo>=</mo><mrow><mo>(</mo></mrow></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> 1 sin θ )cos θ( sin θ cos θ )                    = cos θ sin θ ( sin θ cos θ )                    = sin θcos θ sin θcos θ                   =1
Verifying the Equivalency Using the Even-Odd Identities

Verify the following equivalency using the even-odd identities:

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 1+sin x )[ 1+sin( −x ) ]= cos 2 x

Working on the left side of the equation, we have

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mo stretchy="false">(</mo><mn>1</mn><mo>+</mo><mi>sin</mi><mtext> </mtext><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">[</mo><mn>1</mn><mo>+</mo><mi>sin</mi><mo stretchy="false">(−</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo>=</mo><mo stretchy="false">(</mo><mn>1</mn><mo>+</mo><mi>sin</mi><mtext> </mtext><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>sin</mi><mtext> </mtext><mi>x</mi><mo stretchy="false">)</mo></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> Since sin(−x)=−sin x                                       =1− sin 2 x Difference of squares                                      = cos 2 x cos 2 x=1− sin 2 x
Verifying a Trigonometric Identity Involving sec2θ

Verify the identity<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mfrac/></mrow></annotation-xml></semantics></math> sec 2 θ−1 sec 2 θ = sin 2 θ

As the left side is more complicated, let’s begin there.

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mfrac><mrow><msup><mrow><mi>sec</mi></mrow></msup></mrow></mfrac></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> 2 θ−1 sec 2 θ = ( tan 2 θ+1)−1 sec 2 θ sec 2 θ= tan 2 θ+1                 = tan 2 θ sec 2 θ                 = tan 2 θ( 1 sec 2 θ )                = tan 2 θ( cos 2 θ) cos 2 θ= 1 sec 2 θ                 =( sin 2 θ cos 2 θ )( cos 2 θ) tan 2 θ= sin 2 θ cos 2 θ                 =( sin2 θ cos 2 θ )( cos 2 θ )                 = sin 2 θ

There is more than one way to verify an identity. Here is another possibility. Again, we can start with the left side.

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mfrac><mrow><msup><mrow><mi>sec</mi></mrow></msup></mrow></mfrac></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> 2 θ−1 sec 2 θ = sec 2 θ sec 2 θ − 1 sec 2 θ                  =1− cos 2 θ                  = sin 2 θ
Analysis

In the first method, we used the identity<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msup/></mrow></annotation-xml></semantics></math> sec 2 θ= tan 2 θ+1 and continued to simplify. In the second method, we split the fraction, putting both terms in the numerator over the common denominator. This problem illustrates that there are multiple ways we can verify an identity. Employing some creativity can sometimes simplify a procedure. As long as the substitutions are correct, the answer will be the same.

Show that<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mfrac/></mrow></annotation-xml></semantics></math> cot θ csc θ =cos θ.

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mfrac><mrow><mi>cot</mi><mtext> </mtext><mi>θ</mi></mrow></mfrac></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> csc θ = cos θ sin θ 1 sin θ        = cos θ sin θ ⋅ sin θ 1        =cos θ
Creating and Verifying an Identity

Create an identity for the expression<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>2</mn><mi>tan</mi><mtext> </mtext><mi>θ</mi><mi>sec</mi><mtext> </mtext><mi>θ</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>by rewriting strictly in terms of sine.

There are a number of ways to begin, but here we will use the quotient and reciprocal identities to rewrite the expression:

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mn>2</mn><mtext> </mtext><mi>tan</mi><mtext> </mtext><mi>θ</mi><mi>sec</mi><mtext> </mtext><mi>θ</mi><mo>=</mo><mn>2</mn><mrow><mo>(</mo></mrow></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> sin θ cos θ )( 1 cos θ )                    = 2 sin θ cos 2 θ                    = 2 sin θ 1− sin 2 θ Substitute 1− sin 2  θ for cos 2  θ

Thus,

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>2</mn><mi>tan</mi><mtext> </mtext><mi>θ</mi><mi>sec</mi><mtext> </mtext><mi>θ</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 2 sin θ 1− sin 2  θ
Verifying an Identity Using Algebra and Even/Odd Identities

Verify the identity:

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><msup><mrow><mi>sin</mi></mrow></msup></mrow></mfrac></mrow></annotation-xml></semantics></math> 2 ( −θ )− cos 2 ( −θ ) sin( −θ )−cos( −θ ) =cos θ−sin θ

Let’s start with the left side and simplify:

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mfrac><mrow><msup><mrow><mi>sin</mi></mrow></msup></mrow></mfrac></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> 2 ( −θ )− cos 2 ( −θ ) sin( −θ )−cos( −θ ) = [ sin( −θ ) ] 2 − [ cos( −θ ) ] 2 sin( −θ )−cos( −θ )                                      = (−sin θ ) 2 − ( cos θ ) 2 −sin θ−cos θ sin(−x)=−sin x and cos(−x)=cos x                                      = ( sin θ ) 2 − ( cos θ ) 2 −sin θ−cos θ Difference of squares                                      = ( sin θ−cos θ )( sin θ+cos θ ) −( sin θ+cos θ )                                      = (sin θ−cos θ )( sin θ+cos θ ) −( sin θ+cos θ )                                      =cos θ−sin θ

Verify the identity<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mfrac/></mrow></annotation-xml></semantics></math> sin 2 θ−1 tan θsin θ−tan θ = sin θ+1 tan θ .

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtable><mtr><mtd><mfrac><mrow><msup><mrow><mi>sin</mi></mrow></msup></mrow></mfrac></mtd></mtr></mtable></annotation-xml></semantics></math> 2 θ−1 tan θsin θ−tan θ = ( sin θ+1 )( sin θ−1 ) tan θ( sin θ−1 ) = sin θ+1 tan θ

Verifying an Identity Involving Cosines and Cotangents

Verify the identity:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 1− cos 2 x )( 1+ cot 2 x )=1.

We will work on the left side of the equation.

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><msup/></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> cos 2 x)(1+ cot 2 x)=(1− cos 2 x)( 1+ cos 2 x sin 2 x )                                      =(1− cos 2 x)( sin 2 x sin 2 x + cos 2 x sin2 x )  Find the common denominator.                                      =(1− cos 2 x)( sin 2 x+ cos 2 x sin 2 x )                                     =( sin 2 x)( 1 sin 2 x )                                      =1

Using Algebra to Simplify Trigonometric Expressions

We have seen that algebra is very important in verifying trigonometric identities, but it is just as critical in simplifying trigonometric expressions before solving. Being familiar with the basic properties and formulas of algebra, such as the difference of squares formula, the perfect square formula, or substitution, will simplify the work involved with trigonometric expressions and equations.

For example, the equation<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> sin x+1 )( sin x−1 )=0 resembles the equation<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> x+1 )( x−1 )=0, which uses the factored form of the difference of squares. Using algebra makes finding a solution straightforward and familiar. We can set each factor equal to zero and solve. This is one example of recognizing algebraic patterns in trigonometric expressions or equations.

Another example is the difference of squares formula,<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msup/></mrow></annotation-xml></semantics></math> a 2 − b 2 =( a−b )( a+b ), which is widely used in many areas other than mathematics, such as engineering, architecture, and physics. We can also create our own identities by continually expanding an expression and making the appropriate substitutions. Using algebraic properties and formulas makes many trigonometric equations easier to understand and solve.

Writing the Trigonometric Expression as an Algebraic Expression

Write the following trigonometric expression as an algebraic expression:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>2</mn><msup/></mrow></annotation-xml></semantics></math> cos 2 θ+cos θ−1.

Notice that the pattern displayed has the same form as a standard quadratic expression,<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>a</mi><msup/></mrow></annotation-xml></semantics></math> x 2 +bx+c. Letting<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>cos</mi><mtext> </mtext><mi>θ</mi><mo>=</mo><mi>x</mi><mo>,</mo></mrow></annotation-xml></semantics></math> we can rewrite the expression as follows:

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>2</mn><msup/></mrow></annotation-xml></semantics></math> x 2 +x−1

This expression can be factored as<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 2x+1 )( x−1 ). If it were set equal to zero and we wanted to solve the equation, we would use the zero factor property and solve each factor for<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>x</mi><mo>.</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>At this point, we would replace<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>x</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>with<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>cos</mi><mtext> </mtext><mi>θ</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>and solve for<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>θ</mi><mo>.</mo></mrow></annotation-xml></semantics></math>

Rewriting a Trigonometric Expression Using the Difference of Squares

Rewrite the trigonometric expression:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>4</mn><mtext> </mtext><msup/></mrow></annotation-xml></semantics></math> cos 2 θ−1.

Notice that both the coefficient and the trigonometric expression in the first term are squared, and the square of the number 1 is 1. This is the difference of squares. Thus,

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mn>4</mn><mtext> </mtext><msup/></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> cos 2 θ−1= (2 cos θ) 2 −1                   =(2 cos θ−1)(2 cos θ+1)
Analysis

If this expression were written in the form of an equation set equal to zero, we could solve each factor using the zero factor property. We could also use substitution like we did in the previous problem and let<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>cos</mi><mtext> </mtext><mi>θ</mi><mo>=</mo><mi>x</mi><mo>,</mo></mrow></annotation-xml></semantics></math> rewrite the expression as<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>4</mn><msup/></mrow></annotation-xml></semantics></math> x 2 −1, and factor<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 2x−1 )( 2x+1 ). Then replace<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>x</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>with<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>cos</mi><mtext> </mtext><mi>θ</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>and solve for the angle.

Rewrite the trigonometric expression:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>25</mn><mo>−</mo><mn>9</mn><mtext> </mtext><msup/></mrow></annotation-xml></semantics></math> sin 2  θ.

This is a difference of squares formula:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>25</mn><mo>−</mo><mn>9</mn><mtext> </mtext><msup/></mrow></annotation-xml></semantics></math> sin 2  θ=(5−3 sin θ)(5+3 sin θ).

Simplify by Rewriting and Using Substitution

Simplify the expression by rewriting and using identities:

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msup><mrow><mi>csc</mi></mrow></msup></mrow></annotation-xml></semantics></math> 2 θ− cot 2 θ

We can start with the Pythagorean identity.

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>1</mn><mo>+</mo><msup/></mrow></annotation-xml></semantics></math> cot 2 θ= csc 2 θ

Now we can simplify by substituting<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>1</mn><mo>+</mo><msup/></mrow></annotation-xml></semantics></math> cot 2 θ for<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msup/></mrow></annotation-xml></semantics></math> csc 2 θ. We have

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><msup><mrow><mi>csc</mi></mrow></msup></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> 2 θ− cot 2 θ=1+ cot 2 θ− cot 2 θ                        =1

Use algebraic techniques to verify the identity:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mfrac/></mrow></annotation-xml></semantics></math> cos θ 1+sin θ = 1−sin θ cos θ .

(Hint: Multiply the numerator and denominator on the left side by<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>1</mn><mo>−</mo><mi>sin</mi><mtext> </mtext><mi>θ</mi><mo>.</mo><mo stretchy="false">)</mo></mrow></annotation-xml></semantics></math>

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mfrac><mrow><mi>cos</mi><mtext> </mtext><mi>θ</mi></mrow></mfrac></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> 1+sin θ ( 1−sin θ 1−sin θ )= cos θ(1−sin θ) 1− sin 2 θ                                = cos θ(1−sin θ) cos 2 θ                                =1−sin θ cos θ

Access these online resources for additional instruction and practice with the fundamental trigonometric identities.

Key Equations

Pythagorean identities <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtable columnalign="left"><mtr><mtd><msup><mi>sin</mi></msup></mtd></mtr></mtable></annotation-xml></semantics></math> 2 θ+ cos 2 θ=1 1+ cot 2 θ= csc 2 θ 1+ tan 2 θ= sec 2 θ
Even-odd identities <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtable columnalign="left"><mtr><mtd><mi>tan</mi><mrow><mo>(</mo></mrow></mtd></mtr></mtable></annotation-xml></semantics></math> −θ )=−tan θ cot( −θ )=−cot θ sin( −θ )=−sin θ csc( −θ )=−csc θ cos( −θ )=cos θ sec( −θ )=sec θ
Reciprocal identities <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtable columnalign="left"><mtr><mtd><mi>sin</mi><mtext> </mtext><mi>θ</mi><mo>=</mo><mfrac/></mtd></mtr></mtable></annotation-xml></semantics></math> 1 csc θ cos θ= 1 sec θ tan θ= 1 cot θ csc θ= 1 sin θ sec θ= 1 cos θ cot θ= 1 tan θ
Quotient identities <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow/></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow><mi>tan</mi><mtext> </mtext><mi>θ</mi><mo>=</mo><mfrac/></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> sin θ cos θ cot θ= cos θ sin θ

Key Concepts

  • There are multiple ways to represent a trigonometric expression. Verifying the identities illustrates how expressions can be rewritten to simplify a problem.
  • Graphing both sides of an identity will verify it. See [link].
  • Simplifying one side of the equation to equal the other side is another method for verifying an identity. See [link] and[link].
  • The approach to verifying an identity depends on the nature of the identity. It is often useful to begin on the more complex side of the equation. See [link].
  • We can create an identity by simplifying an expression and then verifying it. See [link].
  • Verifying an identity may involve algebra with the fundamental identities. See [link] and [link].
  • Algebraic techniques can be used to simplify trigonometric expressions. We use algebraic techniques throughout this text, as they consist of the fundamental rules of mathematics. See [link], [link], and [link].

Section Exercises

Verbal

We know<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>cos</mi><mtext> </mtext><mi>x</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>is an even function, and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>sin</mi><mtext> </mtext><mi>x</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>h</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>tan</mi><mtext> </mtext><mi>x</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>are odd functions. What about<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>G</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msup/></mrow></annotation-xml></semantics></math> cos 2 x,F(x)= sin 2 x, and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>H</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msup/></mrow></annotation-xml></semantics></math> tan 2 x? Are they even, odd, or neither? Why?

All three functions,<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>F</mi><mo>,</mo><mi>G</mi><mo>,</mo></mrow></annotation-xml></semantics></math> and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>H</mi><mo>,</mo></mrow></annotation-xml></semantics></math> are even.

This is because<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>F</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> −x )=sin( −x )sin( −x )=( −sin x )( −sin x )= sin 2 x=F( x ),G( −x )=cos( −x )cos( −x )=cos xcos x= cos 2 x=G( x ) and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>H</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> −x )=tan( −x )tan( −x )=( −tan x )( −tan x )= tan 2 x=H( x ).

Examine the graph of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>sec</mi><mtext> </mtext><mi>x</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>on the interval<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mo stretchy="false">[</mo><mo>−</mo><mi>π</mi><mo>,</mo><mi>π</mi><mo stretchy="false">]</mo><mo>.</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>How can we tell whether the function is even or odd by only observing the graph of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>sec</mi><mtext> </mtext><mi>x</mi><mo>?</mo></mrow></annotation-xml></semantics></math>

After examining the reciprocal identity for<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>sec</mi><mtext> </mtext><mi>t</mi><mo>,</mo></mrow></annotation-xml></semantics></math> explain why the function is undefined at certain points.

When<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>cos</mi><mtext> </mtext><mi>t</mi><mo>=</mo><mn>0</mn><mo>,</mo></mrow></annotation-xml></semantics></math> then<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>sec</mi><mtext> </mtext><mi>t</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 1 0 , which is undefined.

All of the Pythagorean identities are related. Describe how to manipulate the equations to get from<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msup/></mrow></annotation-xml></semantics></math> sin 2 t+ cos 2 t=1 to the other forms.

Algebraic

For the following exercises, use the fundamental identities to fully simplify the expression.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>sin</mi><mtext> </mtext><mi>x</mi><mtext> </mtext><mi>cos</mi><mtext> </mtext><mi>x</mi><mtext> </mtext><mi>sec</mi><mtext> </mtext><mi>x</mi></mrow></annotation-xml></semantics></math>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>sin</mi><mtext> </mtext><mi>x</mi></mrow></annotation-xml></semantics></math>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>sin</mi><mo stretchy="false">(</mo><mi>−</mi><mi>x</mi><mo stretchy="false">)</mo><mi>cos</mi><mo stretchy="false">(</mo><mi>−</mi><mi>x</mi><mo stretchy="false">)</mo><mi>csc</mi><mo stretchy="false">(</mo><mi>−</mi><mi>x</mi><mo stretchy="false">)</mo></mrow></annotation-xml></semantics></math>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>tan</mi><mtext> </mtext><mi>x</mi><mi>sin</mi><mtext> </mtext><mi>x</mi><mo>+</mo><mi>sec</mi><mtext> </mtext><mi>x</mi><msup/></mrow></annotation-xml></semantics></math> cos 2 x

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>sec</mi><mtext> </mtext><mi>x</mi></mrow></annotation-xml></semantics></math>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>csc</mi><mtext> </mtext><mi>x</mi><mo>+</mo><mi>cos</mi><mtext> </mtext><mi>x</mi><mi>cot</mi><mo stretchy="false">(</mo><mi>−</mi><mi>x</mi><mo stretchy="false">)</mo></mrow></annotation-xml></semantics></math>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><mi>cot</mi><mtext> </mtext><mi>t</mi><mo>+</mo><mi>tan</mi><mtext> </mtext><mi>t</mi></mrow></mfrac></mrow></annotation-xml></semantics></math> sec(−t)

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>csc</mi><mtext> </mtext><mi>t</mi></mrow></annotation-xml></semantics></math>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>3</mn><mtext> </mtext><msup/></mrow></annotation-xml></semantics></math> sin 3  t csc t+ cos 2  t+2 cos(−t)cos t

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>−</mi><mi>tan</mi><mo stretchy="false">(</mo><mi>−</mi><mi>x</mi><mo stretchy="false">)</mo><mi>cot</mi><mo stretchy="false">(</mo><mi>−</mi><mi>x</mi><mo stretchy="false">)</mo></mrow></annotation-xml></semantics></math>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>−1</mn></mrow></annotation-xml></semantics></math>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><mi>−</mi><mi>sin</mi><mo stretchy="false">(</mo><mi>−</mi><mi>x</mi><mo stretchy="false">)</mo><mi>cos</mi><mtext> </mtext><mi>x</mi><mtext> </mtext><mi>sec</mi><mtext> </mtext><mi>x</mi><mtext> </mtext><mi>csc</mi><mtext> </mtext><mi>x</mi><mtext> </mtext><mi>tan</mi><mtext> </mtext><mi>x</mi></mrow></mfrac></mrow></annotation-xml></semantics></math> cot x

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><mn>1</mn><mo>+</mo><msup/></mrow></mfrac></mrow></annotation-xml></semantics></math> tan 2 θ csc 2 θ + sin 2 θ+ 1 sec 2 θ

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msup><mrow><mi>sec</mi></mrow></msup></mrow></annotation-xml></semantics></math> 2 x

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> tan x csc 2 x + tan x sec 2 x )( 1+tan x 1+cot x )− 1 cos 2 x

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><mn>1</mn><mo>−</mo><msup/></mrow></mfrac></mrow></annotation-xml></semantics></math> cos 2  x tan 2  x +2  sin 2  x

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msup><mrow><mi>sin</mi></mrow></msup></mrow></annotation-xml></semantics></math> 2 x+1

For the following exercises, simplify the first trigonometric expression by writing the simplified form in terms of the second expression.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><mi>tan</mi><mtext> </mtext><mi>x</mi><mo>+</mo><mi>cot</mi><mtext> </mtext><mi>x</mi></mrow></mfrac></mrow></annotation-xml></semantics></math> csc x ; cos x

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><mi>sec</mi><mtext> </mtext><mi>x</mi><mo>+</mo><mi>csc</mi><mtext> </mtext><mi>x</mi></mrow></mfrac></mrow></annotation-xml></semantics></math> 1+tan x ; sin x

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mn>1</mn></mfrac></mrow></annotation-xml></semantics></math> sin x

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><mi>cos</mi><mtext> </mtext><mi>x</mi></mrow></mfrac></mrow></annotation-xml></semantics></math> 1+sin x +tan x; cos x

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mn>1</mn></mfrac></mrow></annotation-xml></semantics></math> sin xcos x −cot x; cot x

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mn>1</mn></mfrac></mrow></annotation-xml></semantics></math> cot x

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mn>1</mn></mfrac></mrow></annotation-xml></semantics></math> 1−cos x − cos x 1+cos x ; csc x

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> sec x+csc x )( sin x+cos x )−2−cot x; tan x

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>tan</mi><mtext> </mtext><mi>x</mi></mrow></annotation-xml></semantics></math>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mn>1</mn></mfrac></mrow></annotation-xml></semantics></math> csc x−sin x ; sec x and tan x

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><mn>1</mn><mo>−</mo><mi>sin</mi><mtext> </mtext><mi>x</mi></mrow></mfrac></mrow></annotation-xml></semantics></math> 1+sin x − 1+sin x 1−sin x ; sec x and tan x

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mo>−</mo><mn>4</mn><mi>sec</mi><mtext> </mtext><mi>x</mi><mi>tan</mi><mtext> </mtext><mi>x</mi></mrow></annotation-xml></semantics></math>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>tan</mi><mtext> </mtext><mi>x</mi><mo>;</mo><mtext> </mtext><mi>sec</mi><mtext> </mtext><mi>x</mi></mrow></annotation-xml></semantics></math>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>sec</mi><mtext> </mtext><mi>x</mi><mo>;</mo><mtext> </mtext><mi>cot</mi><mtext> </mtext><mi>x</mi></mrow></annotation-xml></semantics></math>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mo>±</mo><msqrt/></mrow></annotation-xml></semantics></math> 1 cot 2 x +1

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>sec</mi><mtext> </mtext><mi>x</mi><mo>;</mo><mtext> </mtext><mi>sin</mi><mtext> </mtext><mi>x</mi></mrow></annotation-xml></semantics></math>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>cot</mi><mtext> </mtext><mi>x</mi><mo>;</mo><mtext> </mtext><mi>sin</mi><mtext> </mtext><mi>x</mi></mrow></annotation-xml></semantics></math>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><mo>±</mo><msqrt/></mrow></mfrac></mrow></annotation-xml></semantics></math> 1− sin 2 x sin x

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>cot</mi><mtext> </mtext><mi>x</mi><mo>;</mo><mtext> </mtext><mi>csc</mi><mtext> </mtext><mi>x</mi></mrow></annotation-xml></semantics></math>

For the following exercises, verify the identity.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>cos</mi><mtext> </mtext><mi>x</mi><mo>−</mo><msup/></mrow></annotation-xml></semantics></math> cos 3 x=cos x  sin 2  x

Answers will vary. Sample proof:

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>cos</mi><mtext> </mtext><mi>x</mi><mo>−</mo><msup/></mrow></annotation-xml></semantics></math> cos 3 x=cos x( 1− cos 2 x )

 
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mo>=</mo><mi>cos</mi><mspace width="0.2em"/><mi>x</mi><msup/></mrow></annotation-xml></semantics></math> sin 2 x

 

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>cos</mi><mtext> </mtext><mi>x</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> tan x−sec( −x ) )=sin x−1

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><mn>1</mn><mo>+</mo><msup/></mrow></mfrac></mrow></annotation-xml></semantics></math> sin 2 x cos 2 x = 1 cos 2 x + sin 2 x cos 2 x =1+2  tan 2 x

Answers will vary. Sample proof:

 
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><mn>1</mn><mo>+</mo><msup/></mrow></mfrac></mrow></annotation-xml></semantics></math> sin 2 x cos 2 x = 1 cos 2 x + sin 2 x cos 2 x = sec 2 x+ tan 2 x= tan 2 x+1+ tan 2 x=1+2 tan 2 x

 

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msup><mrow><mrow><mo>(</mo></mrow></mrow></msup></mrow></annotation-xml></semantics></math> sin x+cos x ) 2 =1+2 sin xcos x

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msup><mrow><mi>cos</mi></mrow></msup></mrow></annotation-xml></semantics></math> 2 x− tan 2 x=2− sin 2 x− sec 2 x

Answers will vary. Sample proof:

 
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msup><mrow><mi>cos</mi></mrow></msup></mrow></annotation-xml></semantics></math> 2 x− tan 2 x=1− sin 2 x−( sec 2 x−1 )=1− sin 2 x− sec 2 x+1=2− sin 2 x− sec 2 x

 

Extensions

For the following exercises, prove or disprove the identity.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mn>1</mn></mfrac></mrow></annotation-xml></semantics></math> 1+cos x − 1 1−cos(−x) =−2 cot x csc x

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msup><mrow><mi>csc</mi></mrow></msup></mrow></annotation-xml></semantics></math> 2 x( 1+ sin 2 x )= cot 2 x

False

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> sec 2 (−x)− tan 2 x tan x )( 2+2 tan x 2+2 cot x )−2  sin 2 x=cos 2x

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><mi>tan</mi><mtext> </mtext><mi>x</mi></mrow></mfrac></mrow></annotation-xml></semantics></math> sec x sin( −x )= cos 2 x

False

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><mi>sec</mi><mrow><mo>(</mo></mrow></mrow></mfrac></mrow></annotation-xml></semantics></math> −x ) tan x+cot x =−sin( −x )

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><mn>1</mn><mo>+</mo><mi>sin</mi><mtext> </mtext><mi>x</mi></mrow></mfrac></mrow></annotation-xml></semantics></math> cos x = cos x 1+sin( −x )

Proved with negative and Pythagorean identities

For the following exercises, determine whether the identity is true or false. If false, find an appropriate equivalent expression.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><msup><mrow><mi>cos</mi></mrow></msup></mrow></mfrac></mrow></annotation-xml></semantics></math> 2 θ− sin 2 θ 1− tan 2 θ = sin 2 θ

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>3</mn><mtext> </mtext><msup/></mrow></annotation-xml></semantics></math> sin 2 θ+4  cos 2 θ=3+ cos 2 θ

True<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>3</mn><mtext> </mtext><msup/></mrow></annotation-xml></semantics></math> sin 2 θ+4  cos 2 θ=3  sin 2 θ+3  cos 2 θ+ cos 2 θ=3( sin 2 θ+ cos 2 θ )+ cos 2 θ=3+ cos 2 θ

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><mi>sec</mi><mtext> </mtext><mi>θ</mi><mo>+</mo><mi>tan</mi><mtext> </mtext><mi>θ</mi></mrow></mfrac></mrow></annotation-xml></semantics></math> cot θ+cos θ = sec 2 θ

Glossary

even-odd identities
set of equations involving trigonometric functions such that if<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>f</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> −x )=−f( x ),the identity is odd, and if<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>f</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> −x )=f( x ),the identity is even
Pythagorean identities
set of equations involving trigonometric functions based on the right triangle properties
quotient identities
pair of identities based on the fact that tangent is the ratio of sine and cosine, and cotangent is the ratio of cosine and sine
reciprocal identities
set of equations involving the reciprocals of basic trigonometric definitions