6. Trigonometric Identities and More Equations

For each expression in Column I, use an identity to choose an expression from Column II with the same value. Choices may be used once, more than once, or not at all.

sin 35°

Concept Check Does there exist an angle θ with the function values cos θ = ⅔ and sin θ = ¾?

Factor each trigonometric expression.

sin³ α + cos³ α

Work each problem.

Find the exact values of sin x, cos x, and tan x, for x = π/12 , using

a. difference identities

b. half-angle identities. 

In Exercises 67–74, rewrite each expression in terms of the given function or functions. $\frac{\cos x}{1+\sin x}+\tan x$ ; $\cos x$

Simplify each expression.

±√[(1 - cos 8θ)/(1 + cos 8θ)]

Simplify each expression.

±√[(1 + cos 18x)/2]

Verify that each equation is an identity.

tan (θ/2) = csc θ - cot θ

Verify that each equation is an identity.

csc A sin 2A - sec A = cos 2A sec A

Write each expression in terms of sine and cosine, and then simplify the expression so that no quotients appear and all functions are of θ only. See Example 3.

sin θ sec θ

Be sure that you've familiarized yourself with the second set of formulas presented in this section by working C5–C8 in the Concept and Vocabulary Check. In Exercises 9–22, express each sum or difference as a product. If possible, find this product's exact value. sin 75° + sin 15°

Write each expression in terms of sine and cosine, and then simplify the expression so that no quotients appear and all functions are of θ only. See Example 3.

(sec²θ - 1)/(csc²θ - 1)

Which of the following is not a variation of a Pythagorean identity?

Which of the following is a correct Pythagorean trigonometric identity?

In Exercises 35–38, use the power-reducing formulas to rewrite each expression as an equivalent expression that does not contain powers of trigonometric functions greater than 1. 6 sin⁴ x