KKhaemenes PrecalculusTeacher Answer Key
Student Lesson

Teacher Key · Unit 06 · Lesson 04

Double-Angle and Half-Angle Formulas

Use this key to verify method, notation, domain, units, and interpretation. Accept equivalent exact forms when mathematically valid.

Khaemenes Academy · Teacher Key

Double-Angle and Half-Angle Formulas

U06 · L04
1

Rewrite tan x using sine and cosine.

Identities visualTransform trigonometric expressions using reciprocal, quotient, Pythagorean, angle-sum, double-angle, and half-angle identities.Pθ
Answer: sin x/cos x
Reasoning: Quotient identity.
2

Simplify 1−sin²x.

Identities visualTransform trigonometric expressions using reciprocal, quotient, Pythagorean, angle-sum, double-angle, and half-angle identities.Pθ
Answer: cos²x
Reasoning: Pythagorean identity.
3

Simplify sec²x−tan²x.

Identities visualTransform trigonometric expressions using reciprocal, quotient, Pythagorean, angle-sum, double-angle, and half-angle identities.Pθ
Answer: 1
Reasoning: Rearrange sec²x=1+tan²x.
4

Expand sin(a+b).

Identities visualTransform trigonometric expressions using reciprocal, quotient, Pythagorean, angle-sum, double-angle, and half-angle identities.Pθ
Answer: sin a cos b+cos a sin b
Reasoning: Sine angle-sum identity.
5

Write cos(2x) in terms of cosine only.

Identities visualTransform trigonometric expressions using reciprocal, quotient, Pythagorean, angle-sum, double-angle, and half-angle identities.Pθ
Answer: 2cos²x−1
Reasoning: Double-angle identity.
6

Evaluate sin75° exactly.

Identities visualTransform trigonometric expressions using reciprocal, quotient, Pythagorean, angle-sum, double-angle, and half-angle identities.Pθ
Answer: (√6+√2)/4
Reasoning: Use sin(45°+30°).
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