A-Level Maths / Pure Mathematics / Trigonometry

Reciprocal Trig Functions (sec, cosec, cot)

Definitions, graphs, and identities involving sec, cosec, and cot.

Pure Mathematics A2 50 min

Learning Objectives

Define sec, cosec, and cot in terms of cos, sin, and tan
Sketch the graphs of sec, cosec, and cot, including asymptotes and key features
Use the identities 1 + tan²θ = sec²θ and 1 + cot²θ = cosec²θ
Solve equations involving reciprocal trigonometric functions
Differentiate sec, cosec, and cot using the quotient rule or chain rule

Key Formulae

\sec\theta = \frac{1}{\cos\theta}

\cosec\theta = \frac{1}{\sin\theta}

\cot\theta = \frac{\cos\theta}{\sin\theta} = \frac{1}{\tan\theta}

1 + \tan^2\theta = \sec^2\theta

1 + \cot^2\theta = \cosec^2\theta

\frac{d}{dx}(\sec x) = \sec x \tan x

\frac{d}{dx}(\cosec x) = -\cosec x \cot x

\frac{d}{dx}(\cot x) = -\cosec^2 x

Prior Knowledge Check

Answer at least 3 of 3 correctly to complete this section.

Q1. What is the exact value of

\sin\left(\frac{\pi}{6}\right)

Q2. What is the exact value of

\cos\left(\frac{\pi}{4}\right)

Q3. What is the exact value of

\tan\left(\frac{\pi}{3}\right)

Why This Matters

At AS level you worked with $\sin$ , $\cos$ , and $\tan$ . At A2, you meet their reciprocals: $\sec$ , $\cosec$ , and $\cot$ . These are not new functions — they are shorthand for $\frac{1}{\cos}$ , $\frac{1}{\sin}$ , and $\frac{1}{\tan}$ (or $\frac{\cos}{\sin}$ ). But they come with their own identities, graphs, and derivatives that appear frequently in A2 exam questions.

The two new Pythagorean identities ( $1 + \tan^2\theta = \sec^2\theta$ and $1 + \cot^2\theta = \cosec^2\theta$ ) are particularly important — they are used in integration, differential equations, and proof questions.

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Definitions and Exact Values

Function	Undefined when	Values
$\sec\theta$	$\cos\theta = 0$	$\theta = \frac{\pi}{2}, \frac{3\pi}{2}, \ldots$
$\cosec\theta$	$\sin\theta = 0$	$\theta = 0, \pi, 2\pi, \ldots$
$\cot\theta$	$\sin\theta = 0$ ( $\tan\theta$ undefined)	$\theta = 0, \pi, 2\pi, \ldots$

Angle	$\sec$	$\cosec$	$\cot$
$\frac{\pi}{6}$	$\frac{2}{\sqrt{3}}$	$2$	$\sqrt{3}$
$\frac{\pi}{4}$	$\sqrt{2}$	$\sqrt{2}$	$1$
$\frac{\pi}{3}$	$2$	$\frac{2}{\sqrt{3}}$	$\frac{1}{\sqrt{3}}$

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Identities and Graphs

3/3

Differentiating Reciprocal Trig Functions

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Exam Tips

sec, cosec, and cot are reciprocals of cos, sin, and tan respectively — not inverses (sec is NOT cos⁻¹)
When solving equations, rewrite everything in terms of sin and cos first, then simplify
The identities 1 + tan²θ = sec²θ and 1 + cot²θ = cosec²θ are derived by dividing sin²θ + cos²θ = 1 by cos²θ or sin²θ
For graph sketching, mark the asymptotes first (where the original function equals zero)
When differentiating, write sec x as (cos x)⁻¹ and use the chain rule