Proving identities and using the addition formulae

Identities let you rewrite a trigonometric expression into a form you can simplify or solve. The addition formulae on the formula sheet also give the double-angle results.

Practise this lesson free (6 questions)

Proving an identity

Start from the more complicated side and work towards the other. Never treat it as an equation.
(1 − cos2 θ)(sin θ cos θ) = sin2 θ(sin θ cos θ) = sin θcos θ = tan θ
Use cos2θ + sin2θ = 1 and tan θ = sin θcos θ.

Doubles from the addition formulae

Put B = A in the addition formulae:
sin 2A = 2 sin A cos A
cos 2A = cos2A − sin2A = 1 − 2 sin2A = 2 cos2A − 1
If sin A = 35 and A is acute, cos A = 45, sin 2A = 2425 and cos 2A = 725.

Solving with identities

sin(x + 30°) = 2 cos x: expand, sin x cos 30° + cos x sin 30° = 2 cos x
√32 sin x = 32 cos x, so tan x = √3 and x = 60° or 240° (0° ≤ x < 360°).
2 cos2x + 3 sin x = 3: replace cos2x by 1 − sin2x to get 2 sin2x − 3 sin x + 1 = 0, so sin x = 12 or 1.
Worked example

Solve 2 cos2 x + 3 sin x = 3 for 0° ≤ x ≤ 360°.

  1. 2(1 − sin2 x) + 3 sin x = 3
  2. 2 sin2 x − 3 sin x + 1 = 0, so (2 sin x − 1)(sin x − 1) = 0
  3. sin x = 12: 30°, 150°. sin x = 1: 90°.

Answer: 30°, 90°, 150°

Key idea

Prove identities by working on one side only. sin 2A = 2 sin A cos A and cos 2A = 1 − 2 sin2 A come from the addition formulae. Use cos2 + sin2 = 1 to make an equation in one function.

Quick check

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Extra practice

2 more questions

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  1. Simplify (1 − sin2 θ)(sin θ cos θ).

    1. 1tan θ
    2. 1
    3. tan θ
    4. cos θ
    Show the answer

    Answer: 1tan θ

    cos2 θsin θ cos θ = cos θsin θ = 1tan θ.

  2. sin A = 513 and A is acute. Find sin 2A. Give your answer as a fraction in its simplest form.

    Show the answer

    Answer: 120/169

    cos A = 1213. sin 2A = 2 sin A cos A = 2 × 513 × 1213 = 120169.

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Sources and references

Checked by the GCSE-ready editorial team, 17 October 2026. Last updated 2 October 2026. Spotted a mistake? Tell us and we will fix it.

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