AS & A Level · Practice questions · Secant, cosecant and cotangent

Reciprocals and identities.

Six original Cambridge-style questions on secant, cosecant and cotangent: exact values, graphs, proofs and equations, at Paper 2 and Paper 3 standard.

Original questions All questions on this page are original work, written in the Cambridge AS & A Level style. They are not from past papers. They test the same concepts and skills the syllabus rewards.

The method

Make every term the same function.

Definitions: \(\sec = \dfrac{1}{\cos}\), \(\operatorname{cosec} = \dfrac{1}{\sin}\), \(\cot = \dfrac{\cos}{\sin}\).
Identities: \(\sec^2\theta \equiv 1 + \tan^2\theta\), \(\operatorname{cosec}^2\theta \equiv 1 + \cot^2\theta\).
Equations: substitute so that only one function remains, then solve the quadratic.
Reject impossible roots: \(|\sec\theta| \ge 1\) and \(|\operatorname{cosec}\theta| \ge 1\).
Proofs: start from one side, convert to \(\sin\) and \(\cos\), combine fractions, finish at the other side.

Recall

[3 marks]

Find the exact values of (a) \(\sec 300^\circ\) (b) \(\operatorname{cosec}(-60^\circ)\) (c) \(\cot\dfrac{4\pi}{3}\).

(a) \(\cos 300^\circ = \dfrac{1}{2}\), so \(\sec 300^\circ = 2\). ✓
(b) \(\sin(-60^\circ) = -\dfrac{\sqrt{3}}{2}\), so \(\operatorname{cosec}(-60^\circ) = -\dfrac{2}{\sqrt{3}} = -\dfrac{2\sqrt{3}}{3}\). ✓
(c) \(\tan\dfrac{4\pi}{3} = \tan\dfrac{\pi}{3} = \sqrt{3}\), so \(\cot\dfrac{4\pi}{3} = \dfrac{1}{\sqrt{3}} = \dfrac{\sqrt{3}}{3}\). ✓

Proof

[3 marks]

Prove that \(\tan x + \cot x \equiv \sec x \operatorname{cosec} x\).

LHS \(= \dfrac{\sin x}{\cos x} + \dfrac{\cos x}{\sin x} = \dfrac{\sin^2 x + \cos^2 x}{\sin x \cos x}\). ✓
\(= \dfrac{1}{\sin x\cos x} = \dfrac{1}{\cos x} \times \dfrac{1}{\sin x} = \sec x \operatorname{cosec} x\) = RHS. ✓

Calculation

[4 marks]

Solve the equation \(\sec^2\theta = 3\tan\theta - 1\) for \(0^\circ \le \theta \le 360^\circ\).

\(1 + \tan^2\theta = 3\tan\theta - 1\), so \(\tan^2\theta - 3\tan\theta + 2 = 0\). ✓
\((\tan\theta - 1)(\tan\theta - 2) = 0\), so \(\tan\theta = 1\) or \(\tan\theta = 2\). ✓
\(\tan\theta = 1\): \(\theta = 45^\circ, 225^\circ\). \(\tan\theta = 2\): \(\theta = 63.4^\circ, 243.4^\circ\). ✓
\(\theta = 45^\circ,\ 63.4^\circ,\ 225^\circ,\ 243.4^\circ\). ✓

Calculation

[4 marks]

Solve the equation \(2\cot^2 x + 3\operatorname{cosec} x = 0\) for \(0^\circ < x < 360^\circ\).

\(2(\operatorname{cosec}^2 x - 1) + 3\operatorname{cosec} x = 0\), so \(2\operatorname{cosec}^2 x + 3\operatorname{cosec} x - 2 = 0\). ✓
\((2\operatorname{cosec} x - 1)(\operatorname{cosec} x + 2) = 0\). ✓
\(\operatorname{cosec} x = \dfrac{1}{2}\) would need \(\sin x = 2\): no solutions. ✓
\(\operatorname{cosec} x = -2\): \(\sin x = -\dfrac{1}{2}\), so \(x = 210^\circ, 330^\circ\). ✓

Analysis

[3 marks]

(a) Sketch the graph of \(y = \operatorname{cosec} x\) for \(0^\circ < x < 360^\circ\), showing its asymptotes.

(b) State the set of values of \(k\) for which the equation \(\operatorname{cosec} x = k\) has no solution.

(a) Vertical asymptotes at \(x = 0^\circ, 180^\circ, 360^\circ\). For \(0^\circ < x < 180^\circ\) the curve is U-shaped above the axis with minimum point \((90^\circ, 1)\); for \(180^\circ < x < 360^\circ\) it is an upside-down U below the axis with maximum point \((270^\circ, -1)\). ✓
(b) \(\operatorname{cosec} x\) never lies strictly between \(-1\) and \(1\), so there is no solution for \(-1 < k < 1\). ✓

Analysis

[5 marks]

(a) Show that the equation \(2\operatorname{cosec}^2\theta + 5\cot\theta = 5\) can be written as \(2\cot^2\theta + 5\cot\theta - 3 = 0\).

(b) Hence solve the equation for \(0^\circ < \theta < 180^\circ\).

(a) \(2(1 + \cot^2\theta) + 5\cot\theta = 5\), so \(2\cot^2\theta + 5\cot\theta - 3 = 0\). ✓
(b) \((2\cot\theta - 1)(\cot\theta + 3) = 0\), so \(\cot\theta = \dfrac{1}{2}\) or \(\cot\theta = -3\). ✓
\(\cot\theta = \dfrac{1}{2}\): \(\tan\theta = 2\), \(\theta = 63.4^\circ\). ✓
\(\cot\theta = -3\): \(\tan\theta = -\dfrac{1}{3}\); the calculator gives \(-18.4^\circ\), so in range \(\theta = 180^\circ - 18.4^\circ = 161.6^\circ\). ✓
\(\theta = 63.4^\circ,\ 161.6^\circ\). ✓

Mark this once you have attempted all six and checked your working. It records a Practiced badge on the topic and adds a one-time bonus. Revealing the solutions alone does not count.