6. Trigonometric Identities and More Equations

Verify that each equation is an identity.

sin² β (1 + cot² β) = 1

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

1/( sin α - 1) - 1/(sin α + 1)

Find the remaining five trigonometric functions of θ.

sin θ = -4/5, cos θ < 0

Verify each identity. csc θ - sin θ = cot θ cos θ

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\))$

Simplify the expression.

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

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°

Verify that each equation is an identity.

2 cos² θ - 1 = (1 - tan² θ)/(1 + tan² 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.

cot(-θ)/sec(-θ)

Verify that each equation is an identity.

(2 cot x)/(tan 2x) = csc² x - 2

Verify each identity. cos² θ (1 + tan² θ) = 1

In Exercises 67–74, rewrite each expression in terms of the given function or functions. (sec x + csc x) (sin x + cos x) - 2 - cot x; tan x

Verify that each equation is an identity.

tan² α sin² α = tan² α + cos² α - 1

Match each expression in Column I with its value in Column II.

8. tan (-π/8)

In Exercises 47–54, use the figures to find the exact value of each trigonometric function. sin(θ/2)