Learning Objectives
In this section, you will:
- Verify the fundamental trigonometric identities.
- Simplify trigonometric expressions using algebra and the identities.
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 Table 1), 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 identity 1+cot2θ=csc2θ is found by rewriting the left side of the equation in terms of sine and cosine.
Prove: 1+cot2θ=csc2θ
Similarly, 1+tan2θ=sec2θ can be obtained by rewriting the left side of this identity in terms of sine and cosine. This gives
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 Table 2).
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 which f(−x)= −f(x) for all x in the domain of f. The sine function is an odd function because sin(−θ)=−sinθ. The graph of an odd function is symmetric about the origin. For example, consider corresponding inputs of π2 and −π2. The output of sin(π2) is opposite the output of sin(−π2). Thus,
This is shown in Figure 2.
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 because cos(−θ)=cosθ. For example, consider corresponding inputs π4 and −π4. The output of cos(π4) is the same as the output of cos(−π4). Thus,
See Figure 3.
For all θ in the domain of the sine and cosine functions, respectively, we can state the following:
- Since sin(−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 as tan(−θ)=sin(−θ)cos(−θ)=−sinθcosθ=−tanθ. Tangent is therefore an odd function, which means that tan(−θ)=−tan(θ) for all θ in 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 as cot(−θ)=cos(−θ)sin(−θ)=cosθ−sinθ=−cotθ. Cotangent is therefore an odd function, which means that cot(−θ)=−cot(θ) for all θ 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 csc(−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 as sec(−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 Table 3.
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 Table 4.
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.
Example 1
Graphing the Expressions of an Identity
Graph both sides of the identity cotθ=1tanθ. In other words, on the graphing calculator, graph y=cotθ and y=1tanθ.
Solution
See Figure 4.
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.
Example 2
Verifying a Trigonometric Identity
Verify tanθcosθ=sinθ.
Solution
We will start on the left side, as it is the more complicated side:
Analysis
This identity was fairly simple to verify, as it only required writing tanθ in terms of sinθ and cosθ.
Try It #1
Verify the identity cscθcosθtanθ=1.
Example 3
Verifying the Equivalency Using the Even-Odd Identities
Verify the following equivalency using the even-odd identities:
Solution
Working on the left side of the equation, we have
Example 4
Verifying a Trigonometric Identity Involving sec2θ
Verify the identity sec2θ−1sec2θ=sin2θ
Solution
As the left side is more complicated, let’s begin there.
There is more than one way to verify an identity. Here is another possibility. Again, we can start with the left side.
Analysis
In the first method, we used the identity sec2θ=tan2θ+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.
Try It #2
Show that cotθcscθ=cosθ.
Example 5
Creating and Verifying an Identity
Create an identity for the expression 2tanθsecθ by rewriting strictly in terms of sine.
Solution
There are a number of ways to begin, but here we will use the quotient and reciprocal identities to rewrite the expression:
Thus,
Example 6
Verifying an Identity Using Algebra and Even/Odd Identities
Verify the identity:
Solution
Let’s start with the left side and simplify:
Try It #3
Verify the identity sin2θ−1tanθsinθ−tanθ=sinθ+1tanθ.
Example 7
Verifying an Identity Involving Cosines and Cotangents
Verify the identity: (1−cos2x)(1+cot2x)=1.
Solution
We will work on the left side of the equation.
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)=0 resembles 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.
Example 8
Writing the Trigonometric Expression as an Algebraic Expression
Write the following trigonometric expression as an algebraic expression: 2cos2θ+cosθ−1.
Solution
Notice that the pattern displayed has the same form as a standard quadratic expression, ax2+bx+c. Letting cosθ=x, we can rewrite the expression as follows:
This expression can be factored as (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 x. At this point, we would replace x with cosθ and solve for θ.
Example 9
Rewriting a Trigonometric Expression Using the Difference of Squares
Rewrite the trigonometric expression: 4cos2θ−1.
Solution
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,
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 cosθ=x, rewrite the expression as 4x2−1, and factor (2x−1)(2x+1). Then replace x with cosθ and solve for the angle.
Try It #4
Rewrite the trigonometric expression: 25−9sin2θ.
Example 10
Simplify by Rewriting and Using Substitution
Simplify the expression by rewriting and using identities:
Solution
We can start with the Pythagorean identity.
Now we can simplify by substituting 1+cot2θ for csc2θ. We have
Try It #5
Use algebraic techniques to verify the identity: cosθ1+sinθ=1−sinθcosθ.
(Hint: Multiply the numerator and denominator on the left side by 1−sinθ.)
Media
Access these online resources for additional instruction and practice with the fundamental trigonometric identities.
7.1 Section Exercises
Verbal
We know g(x)=cosx is an even function, and f(x)=sinx and h(x)=tanx are odd functions. What about G(x)=cos2x,F(x)=sin2x, and H(x)=tan2x? Are they even, odd, or neither? Why?
Examine the graph of f(x)=secx on the interval [−π,π]. How can we tell whether the function is even or odd by only observing the graph of f(x)=secx?
After examining the reciprocal identity for sect, explain why the function is undefined at certain points.
All of the Pythagorean Identities are related. Describe how to manipulate the equations to get from sin2t+cos2t=1 to the other forms.
Algebraic
For the following exercises, use the fundamental identities to fully simplify the expression.
sin(−x)cos(−x)csc(−x)
cscx+cosxcot(−x)
3sin3tcsct+cos2t+2cos(−t)cost
−sin(−x)cosxsecxcscxtanxcotx
(tanxcsc2x+tanxsec2x)(1+tanx1+cotx)−1cos2x
For the following exercises, simplify the first trigonometric expression by writing the simplified form in terms of the second expression.
tanx+cotxcscx;cosx
cosx1+sinx+tanx;cosx
11−cosx−cosx1+cosx;cscx
1cscx−sinx;secx and tanx
tanx;secx
secx;sinx
cotx;cscx
For the following exercises, verify the identity.
cosx(tanx−sec(−x))=sinx−1
(sinx+cosx)2=1+2sinxcosx
Extensions
For the following exercises, prove or disprove the identity.
11+cosx−11−cos(−x)=−2cotxcscx
(sec2(−x)−tan2xtanx)(2+2tanx2+2cotx)−2sin2x=cos2x
sec(−x)tanx+cotx=−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
secθ+tanθcotθ+cosθ=sec2θ