A-level · Trigonometry

The sec/cosec/cot identitiesExam-style questions

The two new Pythagorean identities 1 + tan²θ ≡ sec²θ and 1 + cot²θ ≡ cosec²θ — deriving them from sin²+cos²≡1, and using them to prove identities, simplify expressions and turn equations into a solvable quadratic.

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17 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⁡2θ≡sin⁡2θ sec⁡2θ\tan^2\theta \equiv \sin^2\theta\,\sec^2\theta.
    Level 3 · 2 marksTry it
  2. 2
    Prove that (sec⁡θ−tan⁡θ)(sec⁡θ+tan⁡θ)≡1(\sec\theta - \tan\theta)(\sec\theta + \tan\theta) \equiv 1.
    Level 3 · 2 marksTry it
  3. 3
    Solve tan⁡2θ−sec⁡θ=1\tan^2\theta - \sec\theta = 1 for 0≤θ≤360∘0 \le \theta \le 360^\circ.
    Level 3 · 4 marksTry it
  4. 4
    Solve cosec⁡2θ+cot⁡θ=7\operatorname{cosec}^2\theta + \cot\theta = 7 for 0≤θ≤360∘0 \le \theta \le 360^\circ, giving your answers to 1 decimal place.
    Level 3 · 4 marksTry it
  5. 5
    Simplify sec⁡2θ−1sec⁡2θ\dfrac{\sec^2\theta - 1}{\sec^2\theta} to a single trigonometric function.
    Level 3 · 2 marksTry it

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