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Pure · Trigonometry

Chapter 1 · 3

The idea

Proving trigonometric identities

A reliable method for proving trig identities, drawing on the whole toolkit — Pythagorean, reciprocal, addition and double-angle identities — with the common tactics (one side at a time, convert to sin/cos, combine fractions, factorise).

A full journey — read it, play with it, work it, then earn real exam marks. Everything stays on the timeline below.

In this lesson — start anywhere

Pure · Trigonometry

Proving trigonometric identities

A reliable method for proving trig identities, drawing on the whole toolkit — Pythagorean, reciprocal, addition and double-angle identities — with the common tactics (one side at a time, convert to sin/cos, combine fractions, factorise).

Why it works

The question with nothing to solve

Prove that

cos⁡θ1−sin⁡θ+cos⁡θ1+sin⁡θ≡2sec⁡θ.\frac{\cos\theta}{1 - \sin\theta} + \frac{\cos\theta}{1 + \sin\theta} \equiv 2\sec\theta.

Four marks, and not an xx in sight. The triple bar ≡\equiv announces an identity — a claim that the two sides are equal for every value of θ\theta — so there is nothing to find, only something to show. That changes the rules completely, because every equation move you trust is built on knowing the two sides are equal, which is exactly what hasn't been shown yet. This lesson is the method; by the end, the identity above will be a four-line proof.

Keep reading — free

The rest of the explanation, plus 3 worked examples you step through move by move.

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