# Trigonometric Identities and Equations

## Solving Trigonometric Equations with 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 ), 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.
The second and third identities can be obtained by manipulating the first. The identity 
 is found by rewriting the left side of the equation in terms of sine and cosine.
 Prove: 
 
Similarly, 
 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 ).
Recall that an odd function is one in which 
 for all 
 in the domain of 
 The sine function is an odd function because 
 The graph of an odd function is symmetric about the origin. For example, consider corresponding inputs of 
 and 
 The output of 
 is opposite the output of 
 Thus,
This is shown in .
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 

For example, consider corresponding inputs 
 and 
 The output of 
 is the same as the output of 
 Thus,
See .
For all 
 in the domain of the sine and cosine functions, respectively, we can state the following:
1. Since 
 sine is an odd function.
2. Since, 
 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, 
 We can interpret the tangent of a negative angle as 
 Tangent is therefore an odd function, which means that 
 for all 
 in the domain of the tangent function.
The cotangent identity, 
 also follows from the sine and cosine identities. We can interpret the cotangent of a negative angle as 
 Cotangent is therefore an odd function, which means that 
 for all 
 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 
 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 
 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 .
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 .
The reciprocal and quotient identities are derived from the definitions of the basic trigonometric functions.

### 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 
 resembles the equation 
 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, 
 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.

### Key Equations


### Key Concepts

1. There are multiple ways to represent a trigonometric expression. Verifying the identities illustrates how expressions can be rewritten to simplify a problem.
2. Graphing both sides of an identity will verify it. See .
3. Simplifying one side of the equation to equal the other side is another method for verifying an identity. See  and .
4. 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 .
5. We can create an identity by simplifying an expression and then verifying it. See .
6. Verifying an identity may involve algebra with the fundamental identities. See  and .
7. 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 , , and .

### Section Exercises


### Verbal


### Algebraic

For the following exercises, use the fundamental identities to fully simplify the expression.
For the following exercises, simplify the first trigonometric expression by writing the simplified form in terms of the second expression.
For the following exercises, verify the identity.

### Extensions

For the following exercises, prove or disprove the identity.
For the following exercises, determine whether the identity is true or false. If false, find an appropriate equivalent expression.