Proving trig identities

5 minAS Level1.5.4

AI-drafted, independently checked · not yet teacher-reviewed

By the end of this lesson

By the end you can prove a trig identity: start from one side, transform it line by line using the two identities and some algebra, and arrive exactly at the other side.

The rules of the game

A proof is a chain, not an equation to solve. (LHS means the left-hand side of the identity; RHS is the right-hand side.)

Pick the messier side (usually the left) and work on it alone.
Transform it one step at a time, each line identical to the last.
Stop when it looks exactly like the other side, and say so: " $=$ RHS".

What you must not do: write the identity down and start doing things to both sides (cross-multiplying, squaring, moving terms across).

That assumes the thing you're trying to prove is already true.

The toolbox

Move	When to use it
$\tan\theta \to \dfrac{\sin\theta}{\cos\theta}$	any time tan appears; get everything in sin and cos
$\sin^2\theta + \cos^2\theta \to 1$	the moment you see both squares added
$1 - \sin^2\theta \to \cos^2\theta$ (and vice versa)	to swap one square for the other
combine fractions	two fractions added or subtracted
factorise, e.g. $1 - \cos^2\theta = (1 - \cos\theta)(1 + \cos\theta)$	to cancel a bracket in a denominator
expand brackets	squared brackets like $(\sin\theta + \cos\theta)^2$

Worked example: Expand and tidy

Prove that $(\sin\theta + \cos\theta)^2 + (\sin\theta - \cos\theta)^2 \equiv 2$ .

Have a go first?

Quick check

Multiple choiceQuestion 1 · Recall

Answer the 2 questions above to continue.