mathsGrade 11-122 min read

Trigonometric identities worth memorising

The short list of identities that actually earn their keep in exams — plus how each one is really just the Pythagorean theorem in disguise.

There are dozens of trig identities. You need a handful. These are the ones that show up again and again — and almost all of them trace back to a single right-angled triangle.

The three you must know cold

\sin^2\theta + \cos^2\theta = 1

\tan\theta = \frac{\sin\theta}{\cos\theta}

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

That first one is just Pythagoras: on the unit circle a point is $(\cos\theta, \sin\theta)$ , and its distance from the centre is $1$ . Everything else is built from it — divide the first identity through by $\cos^2\theta$ and the third one falls out for free.

Double-angle formulae

\sin 2\theta = 2\sin\theta\cos\theta

\cos 2\theta = \cos^2\theta - \sin^2\theta = 1 - 2\sin^2\theta

The second form of $\cos 2\theta$ is the one to reach for when an integral or equation contains $\sin^2\theta$ — it lets you swap a square for something linear.

When to use which

If you see…	Reach for…
$\sin^2\theta + \cos^2\theta$	replace with $1$
a lone $\tan\theta$ in a fraction	rewrite as $\frac{\sin\theta}{\cos\theta}$
$\sin^2\theta$ inside an integral	the $\cos 2\theta$ identity
$\sin\theta\cos\theta$	half of $\sin 2\theta$

Don't memorise blindly. Sketch the unit circle once at the top of your rough work. If you forget an identity mid-exam, you can rebuild the core three from that picture in under a minute.

Last revised 12 April 2026.