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Mathematics LibreTexts

2.4: Introduction to Proofs in Trigonometry

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Learning Objectives
  • Simplify a trigonometric expression.
  • Rewrite a trigonometric expression in terms of sines and cosines.
  • Simplify an algebraic expression by performing a trigonometric substitution.
  • Determine the plausibility of an identity graphically and understand this does not prove an identity (but can disprove an identity).
  • Prove basic trigonometric identities.
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We have already been introduced to several fundamental identities in Trigonometry. Namely, the Ratio, Reciprocal, and Pythagorean Identities. We then spent most of that section using those identities to help us evaluate trigonometric functions. In this section, we learn how to use those identities to simplify expressions involving trigonometric functions. We then start dipping our toes into proving identities.

Simplifying Expressions Involving Trigonometric Functions

Recall that an algebraic expression (or mathematical expression) is a combination of symbols that are mathematically "well-formed." The mathematical symbols can include numbers (constants), variables, operations (e.g., addition, subtraction, multiplication, etc.), functions, brackets, and other grouping symbols to help determine the order of operations. The one symbol that is always missing from an expression is the equals sign (=).

Reminder: Equations versus Expressions

Equations have equals signs - expressions do not.

You solve equations, but you simplify expressions.

When we simplify an algebraic expression, we obtain a new expression that has the same values as the old one, but is easier to work with. For example, we can apply the Distributive Law and combine like terms to simplify2x(x6)+3(x+2)=2x212x+3x+6=2x29x+6The new expression is equivalent to the old one, that is, the expressions have the same value when we evaluate them at any value of x. For instance, you can check that, at x=3, the expressions 2x(x6)+3(x+2) and 2x29x+6 become2(3)(36)+3(3+2)=6(3)+3(5)=32(3)29(3)+6=1827+6=3To simplify an expression containing trigonometric functions, we treat each function as a single variable. Compare the two calculations below:8xy6xy=2xy8cos(θ)sin(θ)6cos(θ)sin(θ)=2cos(θ)sin(θ)Both calculations are examples of combining like terms. In the second calculation, we treat cos(θ) and sin(θ) as variables, just as we treat x and y in the first calculation.

In Trigonometry, we are often tasked with simplifying expressions involving trigonometric functions. In doing so, we will use many of our skills from Algebra (e.g., simplifying compound rational expressions, factoring, distributing, etc.) in combination with the identities we have recently discovered (along with those we will soon discover). One strategy for simplifying a trigonometric expression is to reduce the number of different trigonometric functions involved. The following example showcases this process using the Ratio Identities.

Example 2.4.1
  1. Simplify: cos(θ)tan(θ)+sin(θ)
  2. Multiply: (cos(θ)+sin(θ))2
  3. Add: cos(θ)sin(θ)+sin(θ)cos(θ)
  4. Simplify: 3tan(A)+4tan(A)2cos(A)
  5. Simplify: 2sin(B)+2sin(B)
Solutions
  1. We apply the Ratio Identities to replace tan(θ) by sin(θ)cos(θ) and obtaincos(θ)tan(θ)+sin(θ)=cos(θ)(sin(θ)cos(θ))+sin(θ)=sin(θ)+sin(θ)=2sin(θ)
  2. The instruction to multiply implies we need to square the binomial and simplify.(cos(θ)+sin(θ))2=cos2(θ)+2cos(θ)sin(θ)+sin2(θ)(distributing)=cos2(θ)+sin2(θ)+2cos(θ)sin(θ)(Commutative Property of Addition)=1+2cos(θ)sin(θ)(Pythagorean Identity)
  3. While we could use the Ratio Identities to make an equivalent expression, the result would be cot(θ)+tan(θ). At that point, we would be stuck. Instead, let's use the Mathematical Mantra and perform some arithmetic before trying our new Trigonometry skills. Specifically, let's get a common denominator and perform the subtraction.cos(θ)sin(θ)+sin(θ)cos(θ)=cos(θ)sin(θ)cos(θ)cos(θ)+sin(θ)cos(θ)sin(θ)sin(θ)(multiplying each fraction by an expressionequivalent to 1 to get common denominators)=cos2(θ)cos(θ)sin(θ)+sin2(θ)cos(θ)sin(θ)=cos2(θ)+sin2(θ)cos(θ)sin(θ)(adding fractions with like denominators)=1cos(θ)sin(θ)(Pythagorean Identity)
  4. Combine like terms.3tan(A)+4tan(A)2cos(A)=7tan(A)2cos(A)Note that tan(A) and cos(A) are not like terms.
  5. Combine like terms.2sin(B)+2sin(B)=2+sin(B)Note that sin(B) means 1sin(B).

It is often easier to know how a trigonometric expression (an expression involving trigonometric functions) will simplify once you try simplification techniques. In Example 2.4.1a, most students new to Trigonometry would likely never have looked at cos(θ)tan(θ)+sin(θ) and thought, "Hey, I bet that simplifies down to something nice... like 2sin(θ)." Luckily, as you move forward in Trigonometry (and mathematics), you develop an intuition for when an expression can be simplified; however, cultivating this intuition takes time and experimentation.

Checkpoint 2.4.1

Simplify each expression.

  1. (1cos(θ))(1+cos(θ))
  2. 2cos(t)4cos(w)sin(w)+3cos(t)2cos(w)
Answers
  1. sin2(θ)
  2. 5cos(t)4cos(w)sin(w)2cos(w)
Caution

In Checkpoint 2.4.1b, note that cos(t) and cos(w) are not like terms. (We can choose values for t and w so that cos(t) and cos(w) have different values.)

Rewriting Trigonometric Expressions

It is often necessary, especially in Calculus, to rewrite a trigonometric expression in terms of a single trigonometric function. To do so, we must use identities.

Example 2.4.2
  1. Rewrite sin(θ)cos2(θ) as an expression involving only sums or differences of powers of sin(θ).
  2. Rewrite cot(θ) in terms of only cos(θ).
Solutions
  1. Using one of the alternate forms of the Pythagorean Identity, we replace cos2(θ) with 1sin2(θ) to getsin(θ)cos2(θ)=sin(θ)(1sin2(θ))(Pythagorean Identity)=sin(θ)sin3(θ)(Distributive Law)
  2.    cot(θ)=cos(θ)sin(θ)(Ratio Identities)=cos(θ)±1cos2(θ)(Pythagorean Identities (alternate form))=±cos(θ)1cos2(θ)

If we graph the original expression from Example 2.4.2a as y1=sin(x)cos2(x) and our resulting equivalent expression as y2=sin(x)sin3(x), we see that they have the same graph, as shown in Figure 2.4.1 below.1

Figure 2.4.1
2.2 Example 5.png

This should convince us that sin(θ)cos2(θ) truly is equivalent to sin(x)sin3(x).

The results of Example 2.4.2 can be thought of as two new identities,sin(θ)cos2(θ)=sin(θ)sin3(θ)andcot(θ)=±cos(θ)1cos2(θ),however, before you get too concerned with having to memorize these as two more identities, let's be clear:

Unless formally stated as a theorem, there is no need to memorize the hundreds of identities we will create, prove, or encounter in Trigonometry.

This means that the only identities you are responsible for memorizing (so far) are the Reciprocal, Ratio, and Pythagorean Identities.

Checkpoint 2.4.2
  1. Rewrite sin2(α)cos2(α) as an expression involving only sums or differences of powers of cos(α).
  2. Verify your identity by graphing.
Answers

cos2(α)cos4(α)

The following example is extremely useful in Calculus II (Integral Calculus).

Example 2.4.3

Simplify the expression3x24as much as possible by substituting 2sec(θ) for x.

Solution
3x24=3(2sec(θ))24(substituting 2sec(θ) for x)=34sec2(θ)4(Laws of Exponents)=34(sec2(θ)1)(factoring)=32sec2(θ)1(simplifying)=32tan2(θ)(Pythagorean Identity: tan2(θ)=sec2(θ)1)=32|tan(θ)|(see comments below)

Our solution to Example 2.4.3 needs some clarification to ensure you understand what happened. Most of the work should be understandable; however, two steps might throw you off.

First, from the Pythagorean Identities, we used the fact that1+tan2(θ)=sec2(θ);however, we modified this identity slightly by subtracting 1 from both sides to gettan2(θ)=sec2(θ)1.The implication of that subtle modification cannot be overstated.

Being comfortable with the available identities and willing to manipulate them as needed will play a critical role in your success in Trigonometry.

The second item that needs our attention is the mathematical equivalencetan2(θ)=|tan(θ)|.A lot of students forget about the absolute values. Let's focus on what is happening here.

Suppose a friend of yours is claiming that the equation x2=x is an identity (this is the same as someone saying tan2(θ)=tan(θ)). As an astute mathematics student, you know that even though the equation is true for all positive values of x, it is false for negative values of x. For example, if x=3, thenx2=(3)2=9=3so x2x. The radical symbol stands for the principle square root - that is, the nonnegative square root. Therefore, the left side of the equation, x2, is never negative. Thus, x2 cannot equal x when x is a negative number. The equation is false for x<0.

One way to see that x2 and x are not equivalent is to compare the graphs of Y1=x2 and Y2=x, shown in Figure 2.4.2 below. You can see that x2 and x do not have the same value for x<0.

Figure 2.4.2
Screen Shot 2022-12-30 at 4.10.45 AM.png

Coming back to the last step in the solution of Example 2.4.3, we should now feel comfortable saying that tan2(θ)tan(θ) and we should be okay with sayingtan2(θ)=|tan(θ)|.

Checking the Plausibility of Identities Graphically

Before jumping into how to rigorously prove a trigonometric identity, let's focus on ways to show that a claimed identity is not an identity.

From the discussion after Example 2.4.3, we can see that, to check whether an equation might be an identity, we can compare graphs of Y1=(left side of the equation) and Y2= (right side of the equation). If the two graphs are identical, it is plausible that the equation is an identity. If the two graphs differ, the equation is not an identity.

Read that last paragraph again.

You cannot use a graph to prove an equation is an identity; however, you can use a graph to demonstrate it is not an identity.

This is crucial to understand. The following example provides some clarity.

Example 2.4.4

Which of the following equations might be identities?

  1. sin(2α)=2sin(α)
  2. cos(x+π180)=cos(x)
  3. cos2(θ2)=1+cos(θ)2
Solutions
  1. Compare the graphs of y1=sin(2x) and y2=2sin(x). The Desmos graphs for both equations are shown in the figure below.
    Figure 2.4.3
    2.2 Example 3a.png
    Because there are two distinct graphs, the expressions sin(2x) and 2sin(x) are not equivalent, and consequently, sin(2α)=2sin(α) is not an identity.
  2. This time we graph y1=cos(x+π180) and y2=cos(x).
    Figure 2.4.4
    2.2 Example 3b.png
    Although the graphs appear identical, when we zoom in, we see that the graphs are, indeed, not the same.
    Figure 2.4.5
    2.2 Example 3c.png
    The graphs are so close together that Desmos' resolution does not distinguish them, but zooming in reveals that they are not identical. Because the two graphs differ, the equation cos(x+π180)=cos(x) is not an identity.
  3. Letting y1=cos2(x2) and y2=1+cos(x)2, we get the following graph from Desmos.
    Figure 2.4.6
    2.3 Example 3c Fixed.png
    These two graphs look identical; however, it is best to zoom in to double-check.
    Figure 2.4.7
    2.3 Example 3c 2.png
    No matter how much we zoom in, the two graphs appear identical. Therefore, we can say that it is plausible that cos2(θ2)=1+cos(θ)2 is an identity.

Example 2.4.4 has a few cautionary tales.

Caution: The Trouble with Technology
  1. Example 2.4.4b illustrates that graphs can be deceiving: even if two graphs look identical, it is always a good idea to zoom in or check some numerical values.
  2. The related equation is not an identity if the two graphs are different.
  3. Graphs cannot be used to prove that an equation is an identity.
Checkpoint 2.4.4

Use graphs to decide which of the following equations might be identities.

  1. cos(2θ)=2cos(θ)
  2. cos(2θ)=cos2(θ)sin2(θ)
  3. cos(θ2)=cos2(θ)
Answers

(b)

Proving Identities

We have proved a trigonometric identity when we show that one trigonometric expression is equivalent to another. In Example 2.4.1a we proved that the equationcos(θ)tan(θ)+sin(θ)=2sin(θ)is an identity; it is valid for all values of θ (as long as the tangent function is defined).

A common strategy for proving an identity is to transform one side of the equation using equivalent expressions until it is identical to the other side. To help us choose the transformations at each step of the proof, we try to match the algebraic form of the final expression.

Example 2.4.5

Prove the identity1+tan2(t)=1cos2(t).

Solution

By manipulating the left side of the equation, we will show that the expression 1+tan2(t) is equivalent to 1cos2(t). First, we use the Ratio Identities to write the expression in terms of sines and cosines:1+tan2(t)=1+(sin(t)cos(t))2=1+sin2(t)cos2(t)Next, we notice that the right side of the proposed identity has only one term, so we combine the terms on the left side. So that the fractions have the same denominator, we write 1 as cos2(t)cos2(t).1+sin2(t)cos2(t)=cos2(t)cos2(t)+sin2(t)cos2(t)=cos2(t)+sin2(t)cos2(t)Finally, we apply the Pythagorean Identity to the numerator.cos2(t)+sin2(t)cos2(t)=1cos2(t)Thus, 1+tan2(t)=1cos2(t), and the identity is proved.

When you write out the proof of an identity, your goal is to transform the expression on one side of the identity into the expression on the other, showing one step of the calculation on each line of your proof. You can justify each step to the right of the calculation. The proof of the identity in the previous example would look like this:LHS=1+tan2(t)=1+(sin(t)cos(t))2(Ratio Identities)=1+sin2(t)cos2(t)(Laws of Exponents)=cos2(t)cos2(t)+sin2(t)cos2(t)(Get a common denominator to combine fractions)=cos2(t)+sin2(t)cos2(t)(adding fractions)=1cos2(t)(Pythagorean Identity)=RHSFor now, we will focus on transforming the left side of an equation into the right side; however, as we move forward in Trigonometry, we will adopt the rule of thumb of transforming the more "complicated" side of the equation into the other side.

Exercise 2.4.6

Prove the identity.(sin(θ)cos(θ))21=2sin(θ)cos(θ)

Solution

LHS=(sin(θ)cos(θ))21=sin2(θ)2cos(θ)sin(θ)+cos2(θ)1(distributing)=sin2(θ)+cos2(θ)12cos(θ)sin(θ)(Commutative Property of Addition)=112cos(θ)sin(θ)(Pythagorean Identity)=2cos(θ)sin(θ)(simplifying)=RHS

Checkpoint 2.4.6

Prove the identity 2cos2(x)1=12sin2(x)

Answer

2cos2(x)1=2(1sin2(x))=22sin2(x)1=12sin2(x)

Before we leave this section, it's important to note that you will be proving many identities in Trigonometry. It takes PRACTICE! We have introduced the idea of a proof early so that we spend a good deal of time sharpening those skills and giving advice on tactics as we move forward. For now, the best advice is to do all the proofs in the homework section - even if your instructor does not assign them.


Footnotes

1 If you tried to graph these functions using your graphing calculator or another graphing technology and didn't get graphs similar to those in Figure 2.4.1, it is likely because your graphing device is in degree mode. The graphs we create in Trigonometry require a mode called radian mode. The meaning of these modes will be explained later.


This page titled 2.4: Introduction to Proofs in Trigonometry is shared under a CC BY-SA 12 license and was authored, remixed, and/or curated by Roy Simpson.

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