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
    Prove that

    tan⁡θ+cot⁡θcosec⁡θ≡sec⁡θθ≠nπ2, n∈Z\frac{\tan\theta + \cot\theta}{\operatorname{cosec}\theta} \equiv \sec\theta \qquad \theta \neq \frac{n\pi}{2},\ n \in \mathbb{Z}

    (3 marks)
    Level 2 · 3 marksTry it
  2. 2
    Prove that cosec⁡θ−sin⁡θ≡cos⁡θcot⁡θ\operatorname{cosec}\theta - \sin\theta \equiv \cos\theta\cot\theta.
    Level 3 · 3 marksTry it
  3. 3
    Prove that sin⁡θ1+cos⁡θ+1+cos⁡θsin⁡θ≡2cosec⁡θ\dfrac{\sin\theta}{1 + \cos\theta} + \dfrac{1 + \cos\theta}{\sin\theta} \equiv 2\operatorname{cosec}\theta.
    Level 3 · 3 marksTry it
  4. 4
    Prove that sec⁡4θ−tan⁡4θ≡sec⁡2θ+tan⁡2θ\sec^4\theta - \tan^4\theta \equiv \sec^2\theta + \tan^2\theta.
    Level 3 · 2 marksTry it
  5. 5
    Prove that cos⁡2θcos⁡θ+sin⁡θ≡cos⁡θ−sin⁡θ\dfrac{\cos 2\theta}{\cos\theta + \sin\theta} \equiv \cos\theta - \sin\theta.
    Level 3 · 2 marksTry it

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