6. Trigonometric Identities and More Equations

Use the Pythagorean identities to rewrite the expression as a single term.

$\left(1+\csc\theta\right)\left(1-\csc\theta\right)$

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

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

Simplify the expression.

$\tan^2\theta-\sec^2\theta+1$

Simplify the expression.

$\frac{\tan\left(-\theta\right)}{\sec\left(-\theta\right)}$

Simplify the expression.

$\left(\frac{\tan^2\theta}{\sin^2\theta}-1\right)\csc^2\left(\theta\right)\cos^2\left(-\theta\right)$

Identify the most helpful first step in verifying the identity.

$\left(\frac{\tan^2\theta}{\sin^2\theta}-1\right)=\sec^2\theta\sin^2\left(-\theta\right)$

Identify the most helpful first step in verifying the identity.

$\sec^3\theta=\sec\theta+\frac{\tan^2\theta}{\cos\theta}$

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.

csc θ - sin θ

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

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

sec x/csc x + csc x/sec x

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.

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

cos β(sec β + csc β)

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.