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Name: ____________________ Pre- Calculus 12 Date: _____________ Chapter 6 – Trigonometric Identities 6.1 B – Reciprocal, Quotient, and Pythagorean Identities Verifying and Proving Trigonometric Identities Warm-up Exercise Simplify by expressing each of the following in terms of a single trigonometric function. Have your trigonometric identity sheet in front of you if necessary. cos 1. sin cos 2 2. sin 2 3. tan sec cos 4. 1 cos2 5. cos 2 1 6. 1 tan2 7. sec 2 1 8. 1 sec2 9. 1 csc 2 10. sin 2 cos2 1 11. csc 2 cot 2 13. 1 sin 2 1 tan 2 14. sec 2 1 csc 2 1 12. sin 2 cos2 tan2 sin2 sin 15. cos cos sin Last day, you discovered that proving identities involved knowing how to work with rational expressions. Today, you will learn a few more “tricks” involved when proving identities involving the first 8 basic identities. Complex Fractions Identities often involve complex fractions. Simplify the following: 1 1 2 3 5 1 6 Use the technique you learned above to prove the following identities: Example 1: 1 cos cos 1 sec Example 2: 1 tan 1 tan 1 cot cot 1 Example 3: sin tan sin sec 1 cos Factoring Recall how to factor a difference of squares a 2 b 2 a trigonometric identity. 4 4 sin cos Example 4: Conjugate Trick Recall how to rationalize a binomial denominator Factoring can be used in the proof of sin cos 2 2 2 1 3 This factor of 1 3 can be loosely called the “conjugate” . This technique can be used in some special trigonometric identity proofs. sin 1 cos Example 5: 1 cos sin Common Denominators Recall how to add two rational expressions with binomial denominators: 1 1 1 a 1 a Obtaining a common denominator is often involved in an identity proof. Example 6: 1 1 1 cos 1 cos 2csc 2 6.1B Verifying and Proving Trigonometric Identities Assignment Copy and prove the following identities: 1. sin sec csc tan 1 2. sec 2 sec 2 sin 2 1 3. sin 4 cos4 2sin 2 1 4. tan 2 1 cot 2 sec 2 5. cot 1 cot 1 tan 6. 1 sec 1 cos sec 1 1 cos sin cos tan 2tan sin cot 7. 1 sin csc 1 1 sin csc 1 8. 9. sin 1 cos 1 cos sin 10. tan sec 1 sec 1 tan 12. 2 2 4sec 2 1 sin 1 sin 14. 1 cos tan sin sin tan sin 11. 13. sin sin 2csc 1 cos 1 cos 1 1 sin 1 sin cos2