6. Trigonometric Identities and More Equations

Simplify each expression. See Example 4.

1 - 2 sin² 22 ½°

Determine whether the positive or negative square root should be selected.

sin (-10°) = ± √[(1 - cos (-20°))/2]

Simplify each expression.

±√[(1 + cos 20α)/2]

Simplify each expression.

±√[(1 - cos 8θ)/(1 + cos 8θ)]

Simplify each expression. See Example 4.

cos² π/8 - 1/2

Simplify each expression.

±√[(1 - cos 5A)/(1 + cos 5A)]

Simplify each expression.

± √[(1 + cos (x/4))/2]

Simplify each expression. See Example 4.

tan 34°/2(1 - tan² 34°)

Graph each expression and use the graph to make a conjecture, predicting what might be an identity. Then verify your conjecture algebraically.

(1 - cos 2x)/sin 2x

Simplify each expression.

±√[(1 - cos (3θ/5))/2]

Simplify each expression. See Example 4.

⅛ sin 29.5° cos 29.5°

Graph each expression and use the graph to make a conjecture, predicting what might be an identity. Then verify your conjecture algebraically.

(cos x sin 2x)/1 + cos 2x)

Verify that each equation is an identity.

cot² (x/2) = (1 + cos x)²/(sin² x)

Simplify each expression. See Example 4.

cos² 2x - sin² 2x

Graph each expression and use the graph to make a conjecture, predicting what might be an identity. Then verify your conjecture algebraically.

csc x - cot x