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
(a) Show that the equation $8\tan^2 x + 3 = 6\sec x$ can be written in the form $5\cos^2 x + 6\cos x - 8 = 0.$

(3 marks)
(b) (i) In this question you must show detailed reasoning.

Hence solve, for $0 < \theta < \tfrac12\pi$ , $8\tan^2 4\theta + 3 = 6\sec 4\theta.$ Give your answers correct to 3 decimal places.

(ii) By replacing $\cos 4\theta$ with its small angle approximation, obtain an estimate of the least positive root of the equation in part (b)(i). Verify that this estimate and your smallest answer to part (b)(i) are the same when both are rounded to 2 decimal places.

(6 marks)
Level 5 · 9 marksTry it