The Pythagorean Identities: sec²θ ≡ 1 + tan²θ and cosec²θ ≡ 1 + cot²θ — CAIE A-Level Mathematics

Introduction

The two Pythagorean identities $\sec^2\theta \equiv 1 + \tan^2\theta$ and $\cosec^2\theta \equiv 1 + \cot^2\theta$ are indispensable tools in Pure Mathematics 3. They extend the familiar $\sin^2\theta + \cos^2\theta \equiv 1$ into the language of the reciprocal trigonometric functions, and the 9709 exam exploits them heavily in two distinct ways:

Simplification and exact evaluation — replacing a combination of reciprocal-trig terms with a simpler equivalent.
Solving equations — converting an equation that mixes two different trig functions (e.g. $\sec^2\theta$ and $\tan\theta$ ) into a single-variable equation that can be solved by factorisation or the quadratic formula.

Marks are regularly lost when candidates either cannot recall which identity to use, or fail to handle all solutions in the given interval. This note equips you to avoid both errors.

Core Concept

Starting from $\sin^2\theta + \cos^2\theta \equiv 1$ , divide every term by $\cos^2\theta$ (valid wherever $\cos\theta \neq 0$ ):

\frac{\sin^2\theta}{\cos^2\theta} + 1 \equiv \frac{1}{\cos^2\theta}

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

Written in the standard form used by 9709:

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

Similarly, divide $\sin^2\theta + \cos^2\theta \equiv 1$ by $\sin^2\theta$ (valid wherever $\sin\theta \neq 0$ ):

1 + \frac{\cos^2\theta}{\sin^2\theta} \equiv \frac{1}{\sin^2\theta}

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

Written in standard form:

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

Key strategic insight: whenever an equation or expression contains two reciprocal-trig functions that are related by one of these identities (e.g. $\sec^2\theta$ alongside $\tan\theta$ , or $\cosec^2\theta$ alongside $\cot\theta$ ), substitute to reduce to a single function, then treat the result as a quadratic.

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Introduction

Core Concept

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