Integration Using Trigonometric Identities — CAIE A-Level Mathematics

Introduction

Many trigonometric integrals — such as $\int \sin^2 x \, dx$ or $\int \cos^2(2x) \, dx$ — cannot be evaluated directly from the standard forms table. However, by applying trigonometric identities, particularly the double-angle formulae, these expressions can be rewritten as standard integrals that are straightforward to evaluate.

This technique is explicitly required by the 9709 syllabus (ref 3.5) and appears regularly in Paper 3 examination questions, both as standalone integration tasks and as part of longer problems involving area, volume of revolution, or differential equations.

Core Concept

The central idea is to replace an integrand that cannot be integrated directly with an equivalent expression that can, using a known trigonometric identity.

The key identities used are the double-angle formulae for cosine, rearranged to express $\sin^2\theta$ and $\cos^2\theta$ in terms of $\cos(2\theta)$ :

\cos(2\theta) = 1 - 2\sin^2\theta \implies \sin^2\theta = \frac{1 - \cos(2\theta)}{2}

\cos(2\theta) = 2\cos^2\theta - 1 \implies \cos^2\theta = \frac{1 + \cos(2\theta)}{2}

These identities lower the power of the trigonometric function from 2 to 1, making the integral tractable. The same principle extends to expressions like $\sin^2(3x)$ or $\cos^2\!\left(\tfrac{x}{2}\right)$ — simply replace $\theta$ with the argument in question.

Other identities that appear in this context include:

$\sin(2\theta) = 2\sin\theta\cos\theta$ , used to integrate products $\sin\theta\cos\theta$ .
$\sin^2\theta + \cos^2\theta = 1$ , used to convert between powers.
$1 + \tan^2\theta = \sec^2\theta$ , used to integrate $\tan^2\theta$ .

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Introduction

Core Concept

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