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).

Pure · Trigonometry

Proving trigonometric identities

Why it works

Proving

\text{LHS} \equiv \text{RHS}

is not solving an equation — you don't yet know the two sides are equal, so you may not move things across the

\equiv

or "cross-multiply". Instead, start from one side (usually the messier one) and transform it, step by step, until it becomes the other side.

The toolkit you can draw on:

\sin^2\theta + \cos^2\theta \equiv 1, \quad 1 + \tan^2\theta \equiv \sec^2\theta, \quad 1 + \cot^2\theta \equiv \operatorname{cosec}^2\theta,

the addition formulae, and the double-angle formulae (including the three forms of

\cos 2\theta

).

Tactics, roughly in order of "try this first":

Convert everything to $\sin$ and $\cos$ — it's the great leveller; almost any

identity yields once both sides are in

\sin/\cos

Combine fractions over a common denominator, then look for a

\sin^2 + \cos^2

to collapse to

1

Factorise — spot a difference of two squares like

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

, or a common factor.

For a double angle, expand $\sin 2\theta$ , $\cos 2\theta$ and choose the

\cos 2\theta

form that cancels best.

Two habits that keep proofs honest: never assume the result and work back to something true (that's circular), and write each line as a genuine equality from the previous one. Finish when the side you're working on literally reads as the other.