An identity is a statement that two trigonometric expressions are equal for every value of the variable. Identities can be verified by manipulating one side of the equation using algebraic substitutions and trigonometric identities until it matches the other side, without moving terms across the equal sign. Practice is important to get better at verifying identities, as each one may require a different approach. Students should keep trying different methods and not get discouraged if it takes time to solve an identity.

1. Verifying Trigonometric Identities

7. Let’s look at some examples!

8. Verifying Trigonometric Identities Now we continue on our journey!

11. Example: and Since both sides are the same, the identity is verified.

14. Establish the following identity: In establishing an identity you should NOT move things from one side of the equal sign to the other. Instead substitute using identities you know and simplifying on one side or the other side or both until both sides match. Let's sub in here using reciprocal identity We often use the Pythagorean Identities solved for either sin 2  or cos 2  . sin 2  + cos 2  = 1 solved for sin 2  is sin 2  = 1 - cos 2  which is our left-hand side so we can substitute. We are done! We've shown the LHS equals the RHS

15. Establish the following identity: Let's sub in here using reciprocal identity and quotient identity Another trick if the denominator is two terms with one term a 1 and the other a sine or cosine, multiply top and bottom of the fraction by the conjugate and then you'll be able to use the Pythagorean Identity on the bottom We worked on LHS and then RHS but never moved things across the = sign combine fractions FOIL denominator

18. Establish the identity

19. Establish the identity

20. Establish the identity