In Exercises 35–38, use the power-reducing formulas to rewrite each expression as an equivalent expression that does not contain powers of trigonometric functions greater than 1. sin² x cos² x

In Exercises 59–68, verify each identity.

cos²(θ/2) = (sec θ + 1)/(2 sec θ)

In Exercises 47–54, use the figures to find the exact value of each trigonometric function. sin(θ/2)

In Exercises 39–46, use a half-angle formula to find the exact value of each expression. tan(7𝝅/8)

In Exercises 47–54, use the figures to find the exact value of each trigonometric function. 2sin(θ/2)cos(θ/2)

In Exercises 39–46, use a half-angle formula to find the exact value of each expression. sin 105°

In Exercises 35–38, use the power-reducing formulas to rewrite each expression as an equivalent expression that does not contain powers of trigonometric functions greater than 1. 6 sin⁴ x

In Exercises 47–54, use the figures to find the exact value of each trigonometric function. tan(θ/2)

In Exercises 47–54, use the figures to find the exact value of each trigonometric function. cos(α/2)

In Exercises 55–58, use the given information to find the exact value of each of the following:

a. sin(α/2)

tan α = 4/3, 180° < α < 270°

In Exercises 55–58, use the given information to find the exact value of each of the following:

a. sin(α/2)

sec α = ﹣3, 𝝅/2 < α < 𝝅

In Exercises 55–58, use the given information to find the exact value of each of the following:

b. cos(α/2)

sec α = ﹣3, 𝝅/2 < α < 𝝅

In Exercises 55–58, use the given information to find the exact value of each of the following:

b. cos(α/2)

tan α = 4/3, 180° < α < 270°

In Exercises 55–58, use the given information to find the exact value of each of the following:

c. tan(α/2)

tan α = 4/3, 180° < α < 270°

In Exercises 55–58, use the given information to find the exact value of each of the following:

c. tan(α/2)

sec α = ﹣3, 𝝅/2 < α < 𝝅

In Exercises 7–14, use the given information to find the exact value of each of the following: c. tan 2θ 15 sin θ = -------- , θ lies in quadrant II. 17

Use the given information to find the exact value of each of the following: $\sin 2\theta$

$\sin \theta =\frac{12}{13},\theta \text{ lies in quadrant II.}$

Use the given information to find the exact value of each of the following: $\cos 2\theta$

$\sin \theta =\frac{12}{13},\theta \text{ lies in quadrant II.}$

Use the given information to find the exact value of each of the following: $\tan 2\theta$

$\sin \theta =\frac{12}{13},\theta \text{ lies in quadrant II.}$

Use the given information to find the exact value of each of the following: $\sin 2\theta$

$\cos \theta =\frac{24}{25},\theta \text{ lies in quadrant IV.}$

Use the given information to find the exact value of each of the following: $\cos 2\theta$

$\cos \theta =\frac{24}{25},\theta \text{ lies in quadrant IV.}$

Use the given information to find the exact value of each of the following: $\tan 2\theta$

$\cos \theta =\frac{24}{25},\theta \text{ lies in quadrant IV.}$

Use the given information to find the exact value of each of the following: sin 2θ

cot θ = 2, θ lies in quadrant III.

Use the given information to find the exact value of each of the following: cos 2θ

cot θ = 2, θ lies in quadrant III.

Use the given information to find the exact value of each of the following: tan 2θ

cot θ = 2, θ lies in quadrant III.

Use the given information to find the exact value of each of the following: sin 2θ

sin θ = ﹣2/3, θ lies in quadrant III.

Use the given information to find the exact value of each of the following: cos 2θ

sin θ = ﹣2/3, θ lies in quadrant III.

Use the given information to find the exact value of each of the following: tan 2θ

sin θ = ﹣2/3, θ lies in quadrant III.

In Exercises 15–22, write each expression as the sine, cosine, or tangent of a double angle. Then find the exact value of the expression. 2 sin 15° cos 15°

Write each expression as the sine, cosine, or tangent of a double angle. Then find the exact value of the expression. cos² 105° - sin² 105°