6. Trigonometric Identities and More Equations

Each expression simplifies to a constant, a single function, or a power of a function. Use fundamental identities to simplify each expression.

csc² t - 1

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.

tan(-θ)/sec θ

Use the given information to find each of the following.

cot θ/2, given tan θ = -(√5)/2 , with 90° < θ < 180°

Verify that each equation is an identity.

(1/2)cot (x/2) - (1/2) tan (x/2) = cot x

For each expression in Column I, choose the expression from Column II that completes an identity. One or both expressions may need to be rewritten.

sec x/csc x

II

A. sin ^2 x/cos ^2 x

B.1/(sec ^2 x)

C. sin (-x)

D. csc ^2 x-cot ^2 x + sin ^2 x

E. tan x

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 7x ﹣ sin 3x

Use the given information to find each of the following.

sin y, given cos 2y = -1/3 , π/2 < y < π

In Exercises 63–68, find the exact value of each expression. Do not use a calculator. cos 12° sin 78° + cos 78° sin 12°

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.

(sec θ - 1) (sec θ + 1)

Verify that each equation is an identity.

(tan² α + 1)/ sec α = sec α

Use the even-odd identities to evaluate the expression.

$-\(\cot\]\left\)(\(\theta\[\right\))\(\cdot\]\sin\[\left\)(-\(\theta\]\right\))$

Verify that each equation is an identity.

[(sec θ - tan θ)² + 1]/(sec θ csc θ - tan θ csc θ) = 2 tan θ

Select the expression with the same value as the given expression.

$\(\sec\[\left\)(-\(\frac{4\pi}{5}\]\right\))$

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. cos 4x + cos 2x

Verify that each equation is an identity.

2 cos² (x/2) tan x = tan x+ sin x