1. Lesson 6.5/9.1 Identities & Proofs
An identity is an equation that is true for all values of the variables for which every term of the equation is defined.

2. Quotient Identities
Tan x = sin 𝑥 cos 𝑥 Cot x = cos 𝑥 sin 𝑥

3. Reciprocal Identities
Sin x = 1 csc 𝑥 Cos x = 1 sec 𝑥 Tan x = 1 cot 𝑥 Csc x = 1 sin 𝑥 Sec x = 1 cos 𝑥 Cot x = 1 tan 𝑥

4. Pythagorean Identities
sin2 x + cos2 x = 1 tan2 x + 1 = sec2 x 1 + cot2 x = csc2 x

5. Use the identities to simplify completely.
cot x sec x
2. tan2 x cos2 x + cos2 x

6. 3. 𝑐𝑠𝑐 2 𝑥− 𝑐𝑜𝑡 2 𝑥 𝑠𝑖𝑛 2 𝑥

7. Prove the following:
4. csc x tan x = sec x

8. 5. cot x sin x = cos x

9. 6. sin 𝑥 tan 𝑥 = cos x
7. cot 𝑥 csc 𝑥 = cos 𝑥

10. 8. (1 – cos2 x) csc x = sin x

11. 9. tan 𝑥+ cot 𝑥 csc 𝑥 = sec 𝑥

12. 10. ( 𝑐𝑜𝑠 2 𝑥−1)( 𝑡𝑎𝑛 2 𝑥+1)=− 𝑡𝑎𝑛 2 𝑥