3.7 Trigonometric Identities

Learning Objectives

In this section, you will:

  • Verify the fundamental trigonometric identities.
  • Simplify trigonometric expressions using algebra and the identities.
Photo of international passports.

Figure 1. International passports and travel documents

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 review some trigonometric identities that we have already seen in earlier sections, create some new ones and show how we can use identities, along with basic tools of algebra, to simplify trigonometric expressions.

Some Fundamental Trigonometric Identities

We have previously discussed 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 Table 1).

Table 1
Even-Odd Identities
[latex]\begin{array}{l}\mathrm{tan}\left(-\theta \right)=-\mathrm{tan}\text{ }\left(\theta\right)\hfill \\ \mathrm{cot}\left(-\theta \right)=-\mathrm{cot}\text{ }\left(\theta\right) \hfill \end{array}[/latex] [latex]\begin{array}{l}\mathrm{sin}\left(-\theta \right)=-\mathrm{sin}\text{ }\left(\theta\right) \hfill \\ \mathrm{csc}\left(-\theta \right)=-\mathrm{csc}\text{ }\left(\theta\right) \hfill \end{array}[/latex] [latex]\begin{array}{l}\mathrm{cos}\left(-\theta \right)=\mathrm{cos}\text{ }\left(\theta\right) \hfill \\ \mathrm{sec}\left(-\theta \right)=\mathrm{sec}\text{ }\left(\theta\right) \hfill \end{array}[/latex]

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 Table 2.

Table 2
Reciprocal Identities
[latex]\mathrm{sin}\text{ }\left(\theta\right) =\frac{1}{\mathrm{csc}\text{ }\left(\theta\right)}[/latex] [latex]\mathrm{csc}\text{ }\left(\theta\right) =\frac{1}{\mathrm{sin}\text{ }\left(\theta\right)} [/latex]
[latex]\mathrm{cos}\text{ }\left(\theta\right) =\frac{1}{\mathrm{sec}\text{ }\left(\theta\right)}[/latex] [latex]\mathrm{sec}\text{ }\left(\theta\right) =\frac{1}{\mathrm{cos}\text{ }\left(\theta\right)}[/latex]
[latex]\mathrm{tan}\text{ }\left(\theta\right) =\frac{1}{\mathrm{cot}\text{ }\left(\theta\right) }[/latex] [latex]\mathrm{cot}\text{ }\left(\theta\right) =\frac{1}{\mathrm{tan}\text{ }\left(\theta\right)}[/latex]

Another 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 Table 3.

Table 3
Quotient Identities
[latex]\mathrm{tan}\left(\theta\right)=\frac{\mathrm{sin}\left(\theta\right) }{\mathrm{cos}\left(\theta\right)}[/latex] [latex]\mathrm{cot}\left(\theta\right)=\frac{\mathrm{cos}\left(\theta\right) }{\mathrm{sin}\left(\theta\right) }[/latex]

Alternate Forms of the Pythagorean Identity

We can use these fundamental identities to derive alternative forms of the Pythagorean Identity,

[latex]{\mathrm{cos}}^{2}\left(\theta\right)+{\mathrm{sin}}^{2}\left(\theta\right)=1.[/latex]

You should recall that the identity shown above is a direct result of our definition of the sine and cosine functions in terms of the coordinates of points on the unit circle.  We can derive two more identities using the methods shown below.

The identity [latex]1+\mathrm{cot}^{2}\left(\theta\right) =\mathrm{csc}^{2}\left(\theta\right) [/latex] is found by dividing each term of the first identity by [latex]\mathrm{sin}^{2}\left(\theta\right)[/latex], and then rewriting each part of the equation using the identities we have already discussed in earlier sections.

[latex]\begin{align*}\frac{\mathrm{sin}^{2}\left(\theta \right)}{\mathrm{sin}^{2}\left(\theta\right)}+\frac{\mathrm{cos}^{2}\left(\theta\right)}{\mathrm{sin}^{2}\left(\theta\right)}& =\frac{1}{\mathrm{sin}^{2}\left(\theta\right)}\end{align*}[/latex]

We can then use our earlier quotient and reciprocal identities to rewrite the expression in this equation as shown below.

[latex]\begin{align*}1+\mathrm{cot}^{2}\left(\theta\right)& =\mathrm{csc}^{2}\left(\theta\right)\end{align*}[/latex]

Similarly, [latex]1+\mathrm{tan}^{2}\left(\theta\right) =\mathrm{sec}^{2}\left(\theta\right) [/latex] can be obtained by dividing all terms in the first identity by [latex]\mathrm{cos}^{2}\left(\theta\right)[/latex], and then rewriting each part of the equation using the quotient and reciprocal identities .

[latex]\begin{align*}\frac{\mathrm{sin}^{2}\left(\theta \right)}{\mathrm{cos}^{2}\left(\theta\right)}+\frac{\mathrm{cos}^{2}\left(\theta\right)}{\mathrm{cos}^{2}\left(\theta\right)}& =\frac{1}{\mathrm{cos}^{2}\left(\theta\right)}\end{align*}[/latex]

[latex]\begin{align*}\mathrm{tan}^{2}\left(\theta\right)+1& =\mathrm{sec}^{2}\left(\theta\right)\end{align*}[/latex]

We now have the three Pythagorean identities shown in Table 4.  

Table 4
Pythagorean Identities
[latex]\mathrm{sin}^{2}\left(\theta\right) +\mathrm{cos}^{2}\left(\theta\right) =1[/latex] [latex]1+{\mathrm{cot}}^{2}\left(\theta\right) ={\mathrm{csc}}^{2}\left(\theta\right) [/latex] [latex]1+{\mathrm{tan}}^{2}\left(\theta\right) ={\mathrm{sec}}^{2}\left(\theta\right) [/latex]

Example 1:  Graphing the Equations of an Identity

Graph both sides of the identity [latex]\mathrm{cot}\left(\theta\right) =\frac{1}{\mathrm{tan}\left(\theta\right)}.[/latex] In other words, on the graphing calculator, graph [latex]y=\mathrm{cot}\left(\theta\right) [/latex] and [latex]y=\frac{1}{\mathrm{tan}\left(\theta\right)}.[/latex]

How To:

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.
  5. Note the values not in the domain of the expression on the left and right as the identity does not hold for those values.

Example 2:  Verifying a Trigonometric Identity

Verify [latex]\mathrm{tan}\left(\theta\right) \mathrm{cos}\left(\theta\right)=\mathrm{sin}\left(\theta\right) .[/latex]

Try it #1

Verify the identity [latex]\mathrm{csc}\left(\theta\right)\text{ } \mathrm{cos}\left(\theta\right)\text{ } \mathrm{tan}\left(\theta\right) =1.[/latex]

Example 3:  Verifying the Equivalency Using the Even-Odd Identities

Verify the following equivalency using the even-odd identities:

[latex]\left(1+\mathrm{sin}\text{ }\left(x\right)\right)\left(1+\mathrm{sin}\left(-x\right)\right)={\mathrm{cos}}^{2}\left(x\right).[/latex]

Example 4:  Verifying a Trigonometric Identity Involving sec2(θ)

Verify the identity [latex]\frac{{\mathrm{sec}}^{2}\left(\theta\right) -1}{{\mathrm{sec}}^{2}\left(\theta\right) }={\mathrm{sin}}^{2}\left(\theta\right). [/latex]

Try it #2

Show that [latex]\frac{\mathrm{cot}\text{ }\left(\theta\right) }{\mathrm{csc}\left(\theta\right)}=\mathrm{cos}\left(\theta\right) .[/latex]

Example 5: Creating and Verifying an Identity

Create an identity for the expression [latex]2\text{ }\mathrm{tan}\left(\theta\right) \mathrm{sec}\left(\theta\right) [/latex] by rewriting strictly in terms of sine.

Example 6: Verifying an Identity Using Algebra and Even/Odd Identities

Verify the identity:

[latex]\frac{\mathrm{sin}^{2}\left(-\theta \right)-\mathrm{cos}^{2}\left(-\theta \right)}{\mathrm{sin}\left(-\theta \right)-\mathrm{cos}\left(-\theta \right)}=\mathrm{cos}\left(\theta \right) -\mathrm{sin}\left(\theta \right) [/latex]

Try it #3

Verify the identity [latex]\frac{\mathrm{sin}^{2}\left(\theta \right) -1}{\mathrm{tan}\left(\theta \right)\mathrm{sin}\left(\theta\right) -\mathrm{tan}\left(\theta\right)}=\frac{\mathrm{sin}\left(\theta\right) +1}{\mathrm{tan}\left(\theta \right)}.[/latex]

Example 7: Verifying an Identity Involving Cosines and Cotangents

Verify the identity: [latex]\left(1-{\mathrm{cos}}^{2}\left(x\right)\right)\left(1+{\mathrm{cot}}^{2}\left(x\right)\right)=1.[/latex]

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 when we are solving equations. 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.

An example is the difference of squares formula, [latex]{a}^{2}-{b}^{2}=\left(a-b\right)\left(a+b\right),[/latex] which is widely used in many areas other than mathematics, such as engineering, architecture, and physics

For example, the expression [latex]\mathrm{sin}^2\left(x\right)-1[/latex] resembles the difference of squares [latex]x^2-1.[/latex] Recognizing that [latex]x^2-1[/latex] can be factored as [latex]\left(x+1\right)\left(x-1\right)[/latex] helps us quickly recognize that  [latex]\mathrm{sin}^2\left(x\right)-1[/latex] can be factored as [latex]\left(\mathrm{sin}\left(x\right)+1\right)\left(\mathrm{sin}\left(x\right)-1\right).[/latex]

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 expressions and equations easier to work with.

Example 8: Writing the Trigonometric Expression as an Algebraic Expression

Write the following trigonometric expression as an algebraic expression: [latex]2\mathrm{cos}^{2}\left(\theta\right) +\mathrm{cos}\left(\theta\right) -1.[/latex]

Example 9: Rewriting a Trigonometric Expression Using the Difference of Squares

Rewrite the trigonometric expression: [latex]4\mathrm{cos}^{2}\left(\theta\right) -1.[/latex]

Try it #4

Rewrite the trigonometric expression: [latex]25-9\text{ }\mathrm{sin}^{2}\left(\theta\right) .[/latex]

Example 10: Simplify by Rewriting and Using Substitution

Simplify the expression by rewriting and using identities:

[latex]\mathrm{csc}^{2}\left(\theta\right) -\mathrm{cot}^{2}\left(\theta\right) [/latex]

Try it #5

Use algebraic techniques to verify the identity: [latex]\frac{\mathrm{cos}\left(\theta\right) }{1+\mathrm{sin}\left(\theta\right) }=\frac{1-\mathrm{sin}\left(\theta\right) }{\mathrm{cos}\left(\theta\right)}.[/latex]

(Hint: Multiply the numerator and denominator on the left side by [latex]1-\mathrm{sin}\left(\theta\right) .)[/latex]

Media

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

Key Equations

Pythagorean identities [latex]\begin{align*}{\mathrm{sin}}^{2}\left(\theta\right) +{\mathrm{cos}}^{2}\left(\theta\right)&=1\\ 1+{\mathrm{cot}}^{2}\left(\theta\right) &={\mathrm{csc}}^{2}\left(\theta\right)\\ 1+{\mathrm{tan}}^{2}\left(\theta\right) &={\mathrm{sec}}^{2}\left(\theta\right) \end{align*}[/latex]
Even-odd identities [latex]\begin{align*}\mathrm{tan}\left(-\theta \right)&=-\mathrm{tan}\left(\theta\right) \\ \mathrm{cot}\left(-\theta \right)&=-\mathrm{cot}\left(\theta\right) \\ \mathrm{sin}\left(-\theta \right)&=-\mathrm{sin}\left(\theta\right)\\ \mathrm{csc}\left(-\theta \right)&=-\mathrm{csc}\left(\theta\right) \\ \mathrm{cos}\left(-\theta \right)&=\mathrm{cos}\left(\theta\right) \\ \mathrm{sec}\left(-\theta \right)&=\mathrm{sec}\left(\theta\right) \end{align*}[/latex]
Reciprocal identities [latex]\begin{align*}\mathrm{sin}\left(\theta\right)&=\frac{1}{\mathrm{csc}\left(\theta\right) }\\ \mathrm{cos}\left(\theta\right)&=\frac{1}{\mathrm{sec}\left(\theta\right)}\\ \mathrm{tan}\left(\theta\right) &=\frac{1}{\mathrm{cot}\left(\theta\right) }\\ \mathrm{csc}\left(\theta\right)&=\frac{1}{\mathrm{sin}\left(\theta\right) }\\ \mathrm{sec}\left(\theta\right) &=\frac{1}{\mathrm{cos}\left(\theta\right)}\\ \mathrm{cot}\left(\theta\right)&=\frac{1}{\mathrm{tan}\left(\theta\right)}\end{align*}[/latex]
Quotient identities [latex]\begin{align*} \mathrm{tan}\left(\theta\right) &=\frac{\mathrm{sin}\left(\theta\right)}{\mathrm{cos}\left(\theta\right) }\\ \mathrm{cot}\left(\theta\right) &=\frac{\mathrm{cos}\left(\theta\right) }{\mathrm{sin}\left(\theta\right) }\end{align*}[/latex]
Shift Identities [latex]\begin{align*}\mathrm{sin}\left(\theta\right)&=\mathrm{cos}\left(\theta-\frac{\pi}{2}\right)\\ \mathrm{cos}\left(\theta\right)&=\mathrm{sin}\left(\theta-\frac{\pi}{2}\right)\end{align*}[/latex]

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.
  • Simplifying one side of the equation to equal the other side is another method for verifying an identity.
  • 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.
  • We can create an identity by simplifying an expression and then verifying it.
  • Verifying an identity may involve algebra with the fundamental identities.
  • 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.

Glossary

even-odd identities
set of equations involving trigonometric functions such that if [latex]f\left(-x\right)=-f\left(x\right),[/latex]the identity is odd, and if [latex]f\left(-x\right)=f\left(x\right),[/latex]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