6. Trigonometric Identities and More Equations

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 = ____

Verify that each equation is an identity.

[(sec θ - tan θ)² + 1]/(sec θ csc θ - tan θ csc θ) = 2 tan θ

Verify that each equation is an identity.

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

(sec θ - 1) (sec θ + 1)

Verify that each equation is an identity.

1/(sec α - tan α) = sec α + tan α

Verify that each equation is an identity.

sec² α - 1 = (sec 2α - 1)/(sec 2α + 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.

1 + cot(-θ)/cot(-θ)

Verify that each equation is an identity.

(csc θ + cot θ)/(tan θ + sin θ) = cot θ csc θ

Verify that each equation is an identity.

sin³ θ = sin θ - cos² θ sin 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.

[1 - sin²(-θ)]/[1 + cot²(-θ)]