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).

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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

    11−sin⁡θ+11+sin⁡θ≡2sec⁡2θ\dfrac{1}{1 - \sin\theta} + \dfrac{1}{1 + \sin\theta} \equiv 2\sec^2\theta

    (3 marks)
    (b) Hence solve, for 0∘⩽θ<360∘0^\circ \leqslant \theta < 360^\circ, the equation

    11−sin⁡θ+11+sin⁡θ=8\dfrac{1}{1 - \sin\theta} + \dfrac{1}{1 + \sin\theta} = 8

    (4 marks)
    Level 4 · 7 marksTry it
  2. 2
    (a) Prove that

    1−cos⁡2xsin⁡2x≡tan⁡x\dfrac{1 - \cos 2x}{\sin 2x} \equiv \tan x, sin⁡2x≠0\quad \sin 2x \ne 0. (4 marks)
    (b) Hence solve, for 0<θ<360∘0 < \theta < 360^\circ, the equation

    1−cos⁡2θsin⁡2θ=3sin⁡θ\dfrac{1 - \cos 2\theta}{\sin 2\theta} = 3\sin\theta,

    giving your answers to 1 decimal place. (5 marks)
    Level 4 · 9 marksTry it
  3. 3
    (a) Prove that

    sec⁡θsec⁡θ−cos⁡θ≡cosec⁡2θ,θ≠nπ2,  n∈Z\dfrac{\sec\theta}{\sec\theta - \cos\theta} \equiv \operatorname{cosec}^2\theta, \qquad \theta \neq \dfrac{n\pi}{2},\; n \in \mathbb{Z}

    (3 marks)
    (b) Hence solve, for 0<θ<2π0 < \theta < 2\pi, the equation

    sec⁡θsec⁡θ−cos⁡θ=4\dfrac{\sec\theta}{\sec\theta - \cos\theta} = 4

    giving your answers in terms of π\pi. (3 marks)
    Level 4 · 6 marksTry it