6. Trigonometric Identities and More Equations

Verify that each equation is an identity.

sin² β (1 + cot² β) = 1

Verify that each equation is an identity.

2 cos³ x - cos x = (cos² x - sin² x)/sec x

Verify that each equation is an identity.

sin² α + tan² α + cos² α = sec² α

Verify that each equation is an identity.

(sin 2x)/(sin x) = 2/sec 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

Verify that each equation is an identity.

(sin² θ)/cos θ = sec θ - cos θ

Verify that each equation is an identity.

(2 tan B)/(sin 2B) = sec² B

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.

csc θ cos θ tan θ

Verify that each equation is an identity.

sec⁴ x - sec² x = tan⁴ x + tan² x

Verify that each equation is an identity.

(2 cot x)/(tan 2x) = csc² x - 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.

sin θ sec θ

Verify that each equation is an identity.

(sec α - tan α)² = (1 - sin α)/(1 + sin α)

Verify that each equation is an identity.

csc A sin 2A - sec A = cos 2A sec A

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.

cot² θ(1 + tan² θ)

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

6. sec² x = ____