6. Trigonometric Identities and More Equations

For each expression in Column I, choose the expression from Column II that completes an identity.

4. cot 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

In Exercises 1–60, verify each identity. cot² t /csc t = csc t - sin t

Perform each transformation. See Example 2.

Write sec x in terms of sin x.

Verify that each equation is an identity.

tan α/sec α = sin α

Use the Pythagorean identities to rewrite the expression with no fraction.

$\(\frac{1}{1-\sec\theta}\)$

Work each problem.

Given tan x = -5⁄4, where π/2< x < π, use the trigonometric identities to find cot x, csc x and sec x.

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

(csc θ sec θ)/cot θ

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. sin² x cos² x

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 cos θ

Perform each indicated operation and simplify the result so that there are no quotients.

sec x/csc x + csc x/sec x

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

cot α sin α

In Exercises 59–68, verify each identity. $\cot \frac{x}{2}=\frac{\sin x}{1-\cos x}$

Use the given information to find each of the following.

sin A/2, given cos A/2 = - 3, 90° < A < 180°

Use identities to write each expression in terms of sin θ and cos θ, and then simplify so that no quotients appear and all functions are of θ only.

csc² θ + sec² θ

Find sinθ.

cot θ = -1/3, θ in quadrant IV