Learning Objectives
In this section, you will:
- Verify the fundamental trigonometric identities.
- Simplify trigonometric expressions using algebra and the identities.

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 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 (Figure)), 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 | ||
---|---|---|
sin2θ+cos2θ=1 | 1+cot2θ=csc2θ | 1+tan2θ=sec2θ |
The second and third identities can be obtained by manipulating the first. The identity1+cot2θ=csc2θis found by rewriting the left side of the equation in terms of sine and cosine.
Prove:1+cot2θ=csc2θ
Similarly,1+tan2θ=sec2θcan be obtained by rewriting the left side of this identity in terms of sine and cosine. This gives
Recall that we determined which trigonometric functions are odd and which are even. 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. (See (Figure)).
Even-Odd Identities | ||
---|---|---|
tan(−θ)=−tanθcot(−θ)=−cotθ | sin(−θ)=−sinθcsc(−θ)=−cscθ | cos(−θ)=cosθsec(−θ)=secθ |
Recall that an odd function is one in whichf(−x)=−f(x)for allxin the domain off.The sine function is an odd function becausesin(−θ)=−sinθ.The graph of an odd function is symmetric about the origin. For example, consider corresponding inputs ofπ2and−π2.The output ofsin(π2)is opposite the output ofsin(−π2).Thus,
This is shown in (Figure).

Figure 2. Graph of y=sinθ
Recall that an even function is one in which
The graph of an even function is symmetric about the y-axis. The cosine function is an even function becausecos(−θ)=cosθ.
For example, consider corresponding inputsπ4and−π4.The output ofcos(π4)is the same as the output ofcos(−π4).Thus,
See (Figure).

Figure 3. Graph of y=cosθ
For allθin the domain of the sine and cosine functions, respectively, we can state the following:
- Sincesin(−θ)=−sinθ,sine is an odd function.
- Since,cos(−θ)=cosθ,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,tan(−θ)=−tanθ.We can interpret the tangent of a negative angle astan(−θ)=sin(−θ)cos(−θ)=−sinθcosθ=−tanθ.Tangent is therefore an odd function, which means thattan(−θ)=−tan(θ)for allθin the domain of the tangent function.
The cotangent identity,cot(−θ)=−cotθ,also follows from the sine and cosine identities. We can interpret the cotangent of a negative angle ascot(−θ)=cos(−θ)sin(−θ)=cosθ−sinθ=−cotθ.Cotangent is therefore an odd function, which means thatcot(−θ)=−cot(θ)for allθin 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 ascsc(−θ)=1sin(−θ)=1−sinθ=−cscθ.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 assec(−θ)=1cos(−θ)=1cosθ=secθ.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 (Figure). Recall that we first encountered these identities when defining trigonometric functions from right angles in Right Angle Trigonometry.
Reciprocal Identities | |
---|---|
sinθ=1cscθ | cscθ=1sinθ |
cosθ=1secθ | secθ=1cosθ |
tanθ=1cotθ | cotθ=1tanθ |
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 (Figure).
Quotient Identities | |
---|---|
tanθ=sinθcosθ | cotθ=cosθsinθ |
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.
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.
The reciprocal identities define reciprocals of the trigonometric functions.
The quotient identities define the relationship among the trigonometric functions.
Graphing the Equations of an Identity
Graph both sides of the identitycotθ=1tanθ.In other words, on the graphing calculator, graphy=cotθandy=1tanθ.
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 confirm an identity verified with analytical means. If both expressions give the same graph, then they are most likely identities.
How To
Given a trigonometric identity, verify that it is true.
- 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.
- Look for opportunities to factor expressions, square a binomial, or add fractions.
- Noting which functions are in the final expression, look for opportunities to use the identities and make the proper substitutions.
- If these steps do not yield the desired result, try converting all terms to sines and cosines.
Verifying a Trigonometric Identity
Verifytanθcosθ=sinθ.
Analysis
This identity was fairly simple to verify, as it only required writingtanθin terms ofsinθandcosθ.
Try It
Verify the identitycscθcosθtanθ=1.
Verifying the Equivalency Using the Even-Odd Identities
Verify the following equivalency using the even-odd identities:
Verifying a Trigonometric Identity Involving sec2θ
Verify the identitysec2θ−1sec2θ=sin2θ
Analysis
In the first method, we used the identitysec2θ=tan2θ+1and 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.
Try It
Show thatcotθcscθ=cosθ.
Creating and Verifying an Identity
Create an identity for the expression2tanθsecθby rewriting strictly in terms of sine.
Verifying an Identity Using Algebra and Even/Odd Identities
Verify the identity:
Try It
Verify the identitysin2θ−1tanθsinθ−tanθ=sinθ+1tanθ.
Verifying an Identity Involving Cosines and Cotangents
Verify the identity:(1−cos2x)(1+cot2x)=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(sinx+1)(sinx−1)=0resembles the equation(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,a2−b2=(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:2cos2θ+cosθ−1.
Rewriting a Trigonometric Expression Using the Difference of Squares
Rewrite the trigonometric expression using the difference of squares:4cos2θ−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 letcosθ=x,rewrite the expression as4x2−1,and factor(2x−1)(2x+1).Then replacexwithcosθand solve for the angle.
Try It
Rewrite the trigonometric expression using the difference of squares:25−9sin2θ.
Simplify by Rewriting and Using Substitution
Simplify the expression by rewriting and using identities:
Try It
Use algebraic techniques to verify the identity:cosθ1+sinθ=1−sinθcosθ.
(Hint: Multiply the numerator and denominator on the left side by1−sinθ.)
Access these online resources for additional instruction and practice with the fundamental trigonometric identities.
Key Equations
Pythagorean identities | cos2θ+sin2θ=11+cot2θ=csc2θ1+tan2θ=sec2θ |
Even-odd identities | tan(−θ)=−tanθcot(−θ)=−cotθsin(−θ)=−sinθcsc(−θ)=−cscθcos(−θ)=cosθsec(−θ)=secθ |
Reciprocal identities | sinθ=1cscθcosθ=1secθtanθ=1cotθcscθ=1sinθsecθ=1cosθcotθ=1tanθ |
Quotient identities | tanθ=sinθcosθcotθ=cosθsinθ |
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 (Figure).
- Simplifying one side of the equation to equal the other side is another method for verifying an identity. See (Figure) and (Figure).
- 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 (Figure).
- We can create an identity and then verify it. See (Figure).
- Verifying an identity may involve algebra with the fundamental identities. See (Figure) and (Figure).
- 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 (Figure), (Figure), and (Figure).
Section Exercises
Verbal
We knowg(x)=cosxis an even function, andf(x)=sinxandh(x)=tanxare odd functions. What aboutG(x)=cos2x,F(x)=sin2x,andH(x)=tan2x?Are they even, odd, or neither? Why?
Examine the graph off(x)=secxon the interval[−π,π].How can we tell whether the function is even or odd by only observing the graph off(x)=secx?
After examining the reciprocal identity forsect,explain why the function is undefined at certain points.
All of the Pythagorean identities are related. Describe how to manipulate the equations to get fromsin2t+cos2t=1to the other forms.
Algebraic
For the following exercises, use the fundamental identities to fully simplify the expression.
sinxcosxsecx
sin(−x)cos(−x)csc(−x)
tanxsinx+secxcos2x
cscx+cosxcot(−x)
cott+tantsec(−t)
3sin3tcsct+cos2t+2cos(−t)cost
−tan(−x)cot(−x)
−sin(−x)cosxsecxcscxtanxcotx
1+tan2θcsc2θ+sin2θ+1sec2θ
(tanxcsc2x+tanxsec2x)(1+tanx1+cotx)−1cos2x
1−cos2xtan2x+2sin2x
For the following exercises, simplify the first trigonometric expression by writing the simplified form in terms of the second expression.
tanx+cotxcscx;cosx
secx+cscx1+tanx;sinx
cosx1+sinx+tanx;cosx
1sinxcosx−cotx;cotx
11−cosx−cosx1+cosx;cscx
(secx+cscx)(sinx+cosx)−2−cotx;tanx
1cscx−sinx;secx and tanx
1−sinx1+sinx−1+sinx1−sinx;secx and tanx
tanx;secx
secx;cotx
secx;sinx
cotx;sinx
cotx;cscx
For the following exercises, verify the identity.
cosx−cos3x=cosxsin2x
cosx(tanx−sec(−x))=sinx−1
1+sin2xcos2x=1cos2x+sin2xcos2x=1+2tan2x
(sinx+cosx)2=1+2sinxcosx
cos2x−tan2x=2−sin2x−sec2x
Extensions
For the following exercises, prove or disprove the identity.
11+cosx−11−cos(−x)=−2cotxcscx
csc2x(1+sin2x)=cot2x
(sec2(−x)−tan2xtanx)(2+2tanx2+2cotx)−2sin2x=cos2x
tanxsecxsin(−x)=cos2x
sec(−x)tanx+cotx=−sin(−x)
1+sinxcosx=cosx1+sin(−x)
For the following exercises, determine whether the identity is true or false. If false, find an appropriate equivalent expression.
cos2θ−sin2θ1−tan2θ=sin2θ
3sin2θ+4cos2θ=3+cos2θ
secθ+tanθcotθ+cosθ=sec2θ
Glossary
- even-odd identities
- set of equations involving trigonometric functions such that iff(−x)=−f(x),the identity is odd, and iff(−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
Candela Citations
- Algebra and Trigonometry. Authored by: Jay Abramson, et. al. Provided by: OpenStax CNX. Located at: http://cnx.org/contents/13ac107a-f15f-49d2-97e8-60ab2e3b519c@11.1. License: CC BY: Attribution. License Terms: Download for free at http://cnx.org/contents/13ac107a-f15f-49d2-97e8-60ab2e3b519c@11.1