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Trigonometric Identities

The formula kit for all of trigonometry: evaluate a product or a power sum with the one identity that collapses it.

Questions in the bank
48
q/paper in 2025–26
0.51
Numeric answer
13%
Notes pages
5

Tier: Long tail

When you’ll see it

A trigonometric expression must be reduced to a single number, or its greatest and least values found, with no equation to solve.

How this chapter is tested

This chapter is the formula kit for the rest of trigonometry. The first pages build it: compound angles with signs fixed by the quadrant, double and triple angles, and exact values at 15°, 18° and 36°. The later pages use it to collapse powers, long products and sums to one number.

Most questions are evaluations: a product like cos A cos 2A cos 4A, a sum of fourth or sixth powers, cos(α + β) from two given ratios. Each has one identity that does the work. Time goes when that identity is missed and the expression is expanded instead. The range questions reduce to a sin θ + b cos θ, or to a quantity such as sin²θ cos²θ with a known range.

Everything here is used again: Trigonometric Equations solves what this chapter simplifies, and Inverse Trigonometric Functions reads its values off the same angle formulas.

The sub-skills

The distinct skills inside the chapter, in the order to learn them.

  • Compound angles

    sin(A ± B), cos(A ± B) and tan(A ± B), with each ratio's sign fixed by its quadrant first.

  • Standard values and multiple angles

    sin 18° = (√5 − 1)/4, cos 36° = (√5 + 1)/4; sin 3θ = 3 sin θ − 4 sin³θ and cos 3θ = 4 cos³θ − 3 cos θ.

  • Powers of sine and cosine

    With p = sin²θ cos²θ, sin⁴θ + cos⁴θ = 1 − 2p and sin⁶θ + cos⁶θ = 1 − 3p.

  • Products and telescoping

    cos A cos 2A … cos 2ⁿ⁻¹A = sin 2ⁿA / (2ⁿ sin A); sin θ sin(60° − θ) sin(60° + θ) = (1/4) sin 3θ; multiply a spaced sum by 2 sin(d/2).

  • Maximum, minimum and range

    a sin θ + b cos θ lies between −√(a² + b²) and √(a² + b²); p = sin²θ cos²θ lies between 0 and 1/4.

Traps to expect

Distractor shapes this chapter reuses. The Traps page covers the ones that cut across chapters.

  • The triangle gives the size, not the sign

    sin x = −3/5 in the third quadrant gives cos x = −4/5, not +4/5. Fix every sign from the quadrant before substituting.

  • sin 18° and cos 36° swapped

    (√5 − 1)/4 is about 0.31 and (√5 + 1)/4 about 0.81. A value above 1/2 cannot be sin 18°.

  • The bound needs one angle

    √(a² + b²) bounds a sin θ + b cos θ only when both terms share the angle; sin θ + cos 2θ is not of that form.

  • p stops at 1/4

    sin²θ cos²θ = (1/4) sin²2θ, so its greatest value is 1/4, not 1.

Learn it before you drill it

This chapter has full teaching notes — foundations, worked examples, self-checks and a mastery check for each page. Read the notes once, then drill page by page below.

Trigonometric Identities notes

Drill every Trigonometric Identities question

48 questions from the bank, across 5 subtopics.

Drill the 48 questions

Drill one subtopic at a time

The 5 subtopics, in teaching order.

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