6. Trigonometric Identities and More Equations

Verify that each equation is an identity.

(cot² t - 1)/(1 + cot² t) = 1 - 2 sin² t

Verify that each equation is an identity.

(sin 3t + sin 2t)/(sin 3t - sin 2t ) = tan (5t/2)/(tan (t/2))

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.

cos θ (cos θ - sec θ)

Verify that each equation is an identity.

tan² α sin² α = tan² α + cos² α - 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.

(sec²θ - 1)/(csc²θ - 1)

Verify that each equation is an identity.

sin θ/(1 - cos θ) - sin θ cos θ/( 1 + cos θ) = csc θ (1 + cos² 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.

tan(-θ)/sec θ

Verify that each equation is an identity.

(1 + sin θ)/(1 - sin θ) - (1 - sin θ)/( 1 + sin θ) = 4 tan θ sec 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.

-sec² (-θ) + sin² (-θ) + cos² (-θ)

Verify that each equation is an identity.

sin θ + cos θ = sin θ/(1 - cot θ) + cos θ/(1 - tan θ)

Let csc x = -3. Find all possible values of (sin x + cos x)/sec x.

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.

(1 + sin x + cos x)² = 2(1 + sin x) (1 + cos x)

Verify that each equation is an identity.

(sec α + csc α) (cos α - sin α) = cot α - tan α

Verify that each equation is an identity.

(1 - cos θ)/(1 + cos θ) = 2 csc² θ - 2 csc θ cot θ - 1