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

DifficultyAll Level 2 Level 3 Level 4 Level 5

1
Write $\sec^2\theta$ in terms of $\tan^2\theta$ , and $\operatorname{cosec}^2\theta$ in terms of $\cot^2\theta$ .
Level 2 · 2 marksTry it
2
Prove that $\tan^2\theta \equiv \sin^2\theta\,\sec^2\theta$ .
Level 3 · 2 marksTry it
3
Prove that $(\sec\theta - \tan\theta)(\sec\theta + \tan\theta) \equiv 1$ .
Level 3 · 2 marksTry it
4
Solve $\tan^2\theta - \sec\theta = 1$ for $0 \le \theta \le 360^\circ$ .
Level 3 · 4 marksTry it
5
Solve $\operatorname{cosec}^2\theta + \cot\theta = 7$ for $0 \le \theta \le 360^\circ$ , giving your answers to 1 decimal place.
Level 3 · 4 marksTry it

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