6. Trigonometric Identities and More Equations

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)]

Verify that each equation is an identity.

sin² θ (1 + cot² θ) - 1 = 0

Verify that each equation is an identity.

2 cos² (x/2) tan x = tan x+ sin 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.

csc θ - sin θ

Verify that each equation is an identity.

(sin⁴ α - cos⁴ α )/(sin² α - cos² α) = 1

Verify that each equation is an identity.

(1/2)cot (x/2) - (1/2) tan (x/2) = cot 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.

(sin θ - cos θ) (csc θ + sec θ)