A-level · Trigonometry

Proving trigonometric identitiesExam-style questions

A reliable method for proving trig identities, drawing on the whole toolkit — Pythagorean, reciprocal, addition and double-angle identities — with the common tactics (one side at a time, convert to sin/cos, combine fractions, factorise).

Start practising Read the lesson

20 questions · A-level level · mark scheme and worked solution on each · no sign-up to try

DifficultyAllLevel 2Level 3Level 4Level 5
  1. 1
    (a) Prove that

    1−cos⁡2θ+sin⁡2θ1+cos⁡2θ+sin⁡2θ≡tan⁡θ\dfrac{1 - \cos 2\theta + \sin 2\theta}{1 + \cos 2\theta + \sin 2\theta} \equiv \tan\theta (4 marks)
    (b) Hence solve the equation

    1−cos⁡4x+sin⁡4x1+cos⁡4x+sin⁡4x=3tan⁡x\dfrac{1 - \cos 4x + \sin 4x}{1 + \cos 4x + \sin 4x} = 3\tan x

    for 0≤x<π0 \le x < \pi, giving your answers in exact form. (5 marks)
    Level 5 · 9 marksTry it
  2. 2
    (a) Prove that

    1sec⁡θ−tan⁡θ≡sec⁡θ+tan⁡θ\dfrac{1}{\sec\theta - \tan\theta} \equiv \sec\theta + \tan\theta

    (3 marks)
    (b) Hence solve the equation

    sec⁡θ−tan⁡θ=3\sec\theta - \tan\theta = 3

    for 0⩽θ⩽2π0 \leqslant \theta \leqslant 2\pi, giving your answer to 33 decimal places. (4 marks)
    Level 5 · 7 marksTry it
  3. 3
    (a) Prove that

    cos⁡3θ≡4cos⁡3θ−3cos⁡θ\cos 3\theta \equiv 4\cos^3\theta - 3\cos\theta

    (4 marks)
    (b) Hence solve, for 0⩽θ<π0 \leqslant \theta < \pi, the equation

    8cos⁡3θ−6cos⁡θ+3=08\cos^3\theta - 6\cos\theta + \sqrt{3} = 0

    giving your answers in exact form in terms of π\pi. (5 marks)
    Level 5 · 9 marksTry it
  4. 4
    (a) Prove that

    sin⁡3x≡3sin⁡x−4sin⁡3x\sin 3x \equiv 3\sin x - 4\sin^3 x. (4 marks)
    (b) Hence solve, for 0≤x<2π0 \le x < 2\pi, the equation

    sin⁡3x=2sin⁡x\sin 3x = 2\sin x,

    giving your answers in exact form. (3 marks)
    (c) Use the identity in part (a) to find the exact value of

    ∫0π3sin⁡3x dx\displaystyle\int_0^{\frac{\pi}{3}} \sin^3 x\,\mathrm{d}x. (2 marks)
    Level 5 · 9 marksTry it
  5. 5
    (a) Prove that

    cot⁡θ−tan⁡θ≡2cot⁡2θ\cot\theta - \tan\theta \equiv 2\cot 2\theta

    (4 marks)
    (b) Hence solve, for 0<θ<π0 < \theta < \pi, the equation

    cot⁡θ−tan⁡θ=23\cot\theta - \tan\theta = 2\sqrt3

    giving your answers in terms of π\pi. (5 marks)
    Level 5 · 9 marksTry it

1 more question on proving trigonometric identities

Free with an account — just an email, no card. The whole bank, every mark scheme and worked solution.

Start free