6. Trigonometric Identities and More Equations

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)

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)