1. Analytic Trigonometry Section 4.1 Trigonometric Identities
Chapter 4
Analytic Trigonometry
Section 4.1
Trigonometric Identities

2. Trigonometric Relations
The six trigonometric functions are related in many different ways. Several of these are quite useful for solving different problems, finding values for the trigonometric functions or solving trigonometric equations.
Basic Trigonometric Identities: (They form the building blocks for many other identities.)
Reciprocal Identities:
Pythagorean Identities
Even/Odd Identities
Cofunction Identities

3. Simplifying Trigonometric Expressions
It is often useful to be able to simplify a trigonometric expression. For example, when you are trying to solve an equation that involves trigonometric functions. There are many ways to do this but the two that are used very often are:
1. Change to sines and cosines.
2. Combine fractions and expressions where possible.
Here are some examples. (Notice these are expressions! (i.e. there is no equal sign).)
Simplify
Simplify
Simplify

4. Verifying Identities
A trigonometric identity is different from a trigonometric expression in that an identity will have an equal (=) sign in it. The point here is to justify by showing the algebra steps why the two sides of the equation are equal. There are many ways (techniques) to go about justifying identities and there is not one that is always easiest, but there are a few that are used often.
1. Start with one side of the equation and see if you can change to be the other side of the equation using algebra steps. (Usually you start with the more complicated side of the equation.)
2. Start with an identity you already know is true and do the same thing to both sides of it to get the identity you want to verify.
3. Change things to sines and cosines to see if you can recognize something in the equation.
Most commonly the first techniques is used. It is often the most straight forward in terms of algebra. It is easiest for us to try to take something that is complicated and make it simpler rather than the other way around.
We will do several examples.

5. Verify:
Verify:
Verify:
Start with the left hand side (LHS).
Start with the left hand side (LHS). Change to sines and cosines
Start with the right hand side (RHS).
Therefore:
Therefore:
Therefore:

6. Verify:
This one is done a bit differently by simplifying both sides of the equation and showing you get equal expressions.
Therefore, since LHS=RHS

7. Trigonometric Substitution
There are times (mostly in calculus) when it becomes necessary to replace the value of x in an expression by a trigonometric function to simplify it and make it easier to deal with.
Example:
Substitute 2cos for x in the expression to the right.
(Assume  is between 0 and /2)