6. Trigonometric Identities and More Equations

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

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

Simplify each expression.

√[(1 + cos 76°)/2]

Simplify each expression.

±√[(1 + cos 20α)/2]

Use a half-angle identity to find each exact value.

sin 195°

Use identities to correctly complete each sentence.

If cos θ = 0.8 and sin θ = 0.6, then tan(-θ) = ________________.