6. Trigonometric Identities and More Equations

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

Verify that each equation is an identity.

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

Express each function as a trigonometric function of x. See Example 5.

cos 3x

Verify that each equation is an identity.

tan (θ/2) = csc θ - cot θ

Express each function as a trigonometric function of x. See Example 5.

cos 4x

Verify that each equation is an identity.

cos x = (1 - tan² (x/2))/(1 + tan² (x/2))

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

2 cos 85° sin 140°

Match each expression in Column I with its value in Column II.

(2 tan (π/3))/(1 - tan² (π/3))

Advanced methods of trigonometry can be used to find the following exact value.

sin 18° = (√5 - 1)/4

(See Hobson's A Treatise on Plane Trigonometry.) Use this value and identities to find each exact value. Support answers with calculator approximations if desired.

cos 18°

Advanced methods of trigonometry can be used to find the following exact value.

sin 18° = (√5 - 1)/4

(See Hobson's A Treatise on Plane Trigonometry.) Use this value and identities to find each exact value. Support answers with calculator approximations if desired.

cot 18°