3.1: Fundamental Identities

Learning Objectives

• Use the fundamental identities to prove other identities.
• Apply the fundamental identities to values of \(\theta\) and show that they are true.

Basic Trigonometric Identities

The basic trigonometric identities are ones that can be logically deduced from the definitions and graphs of the six trigonometric functions. Previously, some of these identities have been used in a casual way, but now they will be formalized and added to the toolbox of trigonometric identities.

Trigonometric Identities

An identity is a mathematical sentence involving the symbol “=” that is always true for variables within the domains of the expressions on either side.

Reciprocal Identities

The reciprocal identities refer to the connections between the trigonometric functions like sine and cosecant. Sine is opposite over hypotenuse and cosecant is hypotenuse over opposite. This logic produces the following six identities.

• \(\sin\theta =\dfrac{1}{\csc\theta}\)
• \(\cos\theta =\dfrac{1}{\sec\theta}\)
• \(\tan\theta =\dfrac{1}{\cot\theta}\)
• \(\cot\theta =\dfrac{1}{\tan\theta}\)
• \(\sec\theta =\dfrac{1}{\cos\theta}\)
• \(\csc\theta =\dfrac{1}{\sin\theta}\)

Quotient Identities

The quotient identities follow from the definition of sine, cosine and tangent.

• \(\tan\theta = \dfrac{\sin\theta}{\cos\theta}\)
• \(\cot\theta =\dfrac{\cos\theta}{\sin\theta}\)

Odd/Even Identities

An even function is a function where the value of the function acting on an argument is the same as the value of the function when acting on the negative of the argument. Or, in short:

\(f(x)=f(−x)\)

So, for example, if \(f(x)\) is some function that is even, then \(f(2)\) has the same answer as \(f(-2)\). \(f(5)\) has the same answer as \(f(-5)\), and so on.

In contrast, an odd function is a function where the negative of the function's answer is the same as the function acting on the negative argument. In math terms, this is:

\(−f(x)=f(−x)\)

If a function were negative, then \(f(-2) = -f(2)\), \(f(-5) = -f(5)\), and so on.

Functions are even or odd depending on how the end behavior of the graphical representation looks. For example, \(y=x^2\) is considered an even function because the ends of the parabola both point in the same direction and the parabola is symmetric about the \(y\)−axis. \(y=x^3\) is considered an odd function for the opposite reason. The ends of a cubic function point in opposite directions and therefore the parabola is not symmetric about the \(y\)−axis. What about the trig functions? They do not have exponents to give us the even or odd clue (when the degree is even, a function is even, when the degree is odd, a function is odd).

\(\dfrac{\text { Even Function }}{y=(-x)^{2}=x^{2}} \quad \dfrac{\text { Odd Function }}{y=(-x)^{3}=-x^{3}}\)

Let’s consider sine. Start with \(\sin(−x)\). Will it equal \(\sin x\) or \(−\sin x\)? Plug in a couple of values to see.

\(\begin{aligned} \sin(−30^{\circ} )&=\sin 330^{\circ} =−\dfrac{1}{2}=−\sin 30^{\circ} \\ \sin(−135^{\circ} ) &=\sin 225^{\circ} =−\dfrac{\sqrt{2}}{2}=−\sin 135^{\circ}\end{aligned}\)

From this we see that sine is odd. Therefore, \(\sin(−x)=−\sin x\), for any value of \(x\). For cosine, we will plug in a couple of values to determine if it’s even or odd.

\(\begin{aligned} \cos(−30^{\circ} )&=\cos 330^{\circ} =\dfrac{\sqrt{3}}{2}=\cos 30^{\circ} \\ \cos(−135^{\circ} ) &=\cos 225^{\circ} =−\dfrac{\sqrt{2}}{2}=\cos 135^{\circ}\end{aligned}\)

This tells us that the cosine is even. Therefore, \(\cos(−x)= \cos x\), for any value of \(x\). The other four trigonometric functions are as follows:

\(\begin{aligned} \tan(−x)&=−\tan x \\ \csc(−x)&=−\csc x \\ \sec(−x)&=\sec x \\ \cot(−x)&=−\cot x \end{aligned}\)

Notice that cosecant is odd like sine and secant is even like cosine.

The odd-even identities follow from the fact that only cosine and its reciprocal secant are even and the rest of the trigonometric functions are odd.

• \(\sin(−\theta )=−\sin\theta\)
• \(\cos(−\theta )=\cos\theta\)
• \(\tan(−\theta )=−\tan\theta\)
• \(\cot(−\theta )=−\cot\theta\)
• \(\sec(−\theta )=\sec\theta\)
• \(\csc(−\theta )=−\csc\theta\)

Cofunction Identities

The cofunction identities make the connection between trigonometric functions and their “co” counterparts like sine and cosine. Graphically, all of the cofunctions are reflections and horizontal shifts of each other.

• \(\cos(\dfrac{\pi}{2}−\theta )=\sin\theta\)
• \(\sin(\dfrac{\pi}{2}−\theta )=\cos\theta\)
• \(\tan(\dfrac{\pi}{2}−\theta )=\cot\theta\)
• \(\cot(\dfrac{\pi}{2}−\theta )=\tan\theta\)
• \(\sec(\dfrac{\pi}{2}−\theta )=\csc\theta\)
• \(\csc(\dfrac{\pi}{2}−\theta )=\sec\theta\)

Example \(\PageIndex{1}\)

Earlier, you were asked how you could simplify the trigonometric expression:

\(\left(\dfrac{\sin(\dfrac{\pi}{2}−\theta )}{\sin(−\theta )}\right)^{−1}\)

Solution

It can be simplified to be equivalent to negative tangent as shown below:

\(\begin{aligned}\left(\dfrac{\sin(\dfrac{\pi}{2}−\theta )}{\sin(−\theta )}\right)^{−1}&=\dfrac{\sin(−\theta )}{\sin(\dfrac{\pi}{2}−\theta )} \\&=−\sin\theta \cos\theta \\&=−\tan\theta\end{aligned}\)

Example \(\PageIndex{2}\)

If \(\sin\theta =0.87\), find \(\cos(\theta −\dfrac{\pi}{2})\).

Solution

While it is possible to use a calculator to find \theta , using identities works very well too.

First you should factor out the negative from the argument. Next you should note that cosine is even and apply the odd-even identity to discard the negative in the argument. Lastly recognize the cofunction identity.

\(\cos(\theta −\dfrac{\pi}{2})=\cos(−(\dfrac{\pi}{2}−\theta ))=\cos(\dfrac{\pi}{2}−\theta )=\sin\theta =0.87\)

Example \(\PageIndex{3}\)

If \(\cos(\theta −\dfrac{\pi}{2})=0.68\) then determine \(\csc(−\theta )\).

Solution

You need to show that \(\cos(\theta −\dfrac{\pi}{2})=\cos(\dfrac{\pi}{2}−\theta )\).

\(\begin{aligned} 0.68&=\cos(\theta −\dfrac{\pi}{2})\\&=\cos(\dfrac{\pi}{2}−\theta ) \\&=\sin(\theta ) \end{aligned}\)

Then, \(\csc(−\theta )=−\csc \theta\)

\(\begin{aligned} &=−\dfrac{1}{\sin\theta} \\ &=−(0.68)^{−1} &\approx −1.47\end{aligned}\)

Example \(\PageIndex{4}\)

Use identities to prove the following: \(\cot(−\beta ) \cot(\dfrac{\pi}{2}−\beta ) \sin(−\beta )= \cos(\beta −\dfrac{\pi}{2})\).

Solution

When doing trigonometric proofs, it is vital that you start on one side and only work with that side until you derive what is on the other side. Sometimes it may be helpful to work from both sides and find where the two sides meet, but this work is not considered a proof. You will have to rewrite your steps so they follow from only one side. In this case, work with the left side and keep rewriting it until you have \(\cos(\beta −\dfrac{\pi}{2})\).

\(\begin{aligned} \cot(−\beta ) \cot(\dfrac{\pi}{2}−\beta ) \sin(−\beta ) \\ &=−\cot \beta \cdot \tan\beta \cdot −\sin\beta \\&=−1\cdot −sin\beta \\&=sin\beta \\&=cos(\dfrac{\pi}{2}−\beta ) \\&=cos(−(\beta −\dfrac{\pi}{2})) \\&=cos(\beta −\dfrac{\pi}{2}) \end{aligned}\)

Example \(\PageIndex{5}\)

Prove the following trigonometric identity by working with only one side.

\(\cos x\sin x\tan x\cot x\sec x\csc x=1\)

Solution

\(\begin{aligned} \cos x\sin x\tan x\cot x\sec x\csc x &=\cos x\sin x\tan x\cdot \dfrac{1}{\tan x} \cdot \dfrac{1}{\cos x}\cdot \dfrac{1}{\sin x} &=1\end{aligned}\)

Review

1. Prove that \(\tan\theta \cdot cot\theta =1\).
2. Prove that \(\sin\theta \cdot \csc\theta =1\).
3. Prove that \(\sin\theta \cdot sec\theta =\tan\theta \).
4. Prove that \(\cos\theta \cdot \csc\theta =\cot\theta \).
5. If \(\sin\theta =0.81\), what is \(\sin(−\theta )\)?
6. If \(\cos\theta =0.5\), what is \(\cos(−\theta )\)?
7. If \(\cos\theta =0.25\), what is \(\sec(−\theta )\)?
8. If \(\csc\theta =0.7\), what is \(\sin(−\theta )\)?

Vocabulary

Term	Definition
cofunction	Cofunctions are functions that are identical except for a reflection and horizontal shift. Examples include: sine and cosine, tangent and cotangent, secant and cosecant.
even	An even function is a function with a graph that is symmetric with respect to the y-axis and has the property that \(f(−x)=f(x)\).
identity	An identity is a mathematical sentence involving the symbol “=” that is always true for variables within the domains of the expressions on either side.
Odd Function	An odd function is a function with the property that \(f(−x)=−f(x)\). Odd functions have rotational symmetry about the origin.
proof	A proof is a series of true statements leading to the acceptance of truth of a more complex statement.

Additional Resources

Interactive Element

Video: Trigonometric Identities - Overview

Practice: Fundamental Trigonometric Identities