6. Trigonometric Identities and More Equations

Verify that each equation is an identity.

(sin 2x)/(2sin x) = cos² (x/2) - sin² (x/2)

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: cos 2θ

sin θ = ﹣2/3, θ lies in quadrant III.

Write each expression as a sum or difference of trigonometric functions. See Example 7.

5 cos 3x cos 2x

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°

Solve each equation in x over the interval [0, 2π) and each equation in θ over the interval [0°, 360°). Give exact solutions.

sin x = sin 2x

Solve each equation (x in radians and θ in degrees) for all exact solutions where appropriate. Round approximate answers in radians to four decimal places and approximate answers in degrees to the nearest tenth. Write answers using the least possible nonnegative angle measures.

2 sin θ = 2 cos 2θ

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°

Solve each equation in x over the interval [0, 2π) and each equation in θ over the interval [0°, 360°). Give exact solutions.

cos 2x + cos x = 0

Write each expression as a sum or difference of trigonometric functions. See Example 7.

8 sin 7x sin 9x

Simplify each expression. See Example 4.

⅛ sin 29.5° cos 29.5°

Write each expression as the sine, cosine, or tangent of a double angle. Then find the exact value of the expression. 2cos² 𝝅/8﹣ 1

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

Simplify each expression. See Example 4.

1 - 2 sin² 22 ½°