TeachingApproachTutoringBlogNotesAboutGet in touch
← All notes
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⁡2θ+cos⁡2θ=1\sin^2\theta + \cos^2\theta = 1 tan⁡θ=sin⁡θcos⁡θ\tan\theta = \frac{\sin\theta}{\cos\theta} 1+tan⁡2θ=sec⁡2θ1 + \tan^2\theta = \sec^2\theta

That first one is just Pythagoras: on the unit circle a point is (cos⁡θ,sin⁡θ)(\cos\theta, \sin\theta), and its distance from the centre is 11. Everything else is built from it — divide the first identity through by cos⁡2θ\cos^2\theta and the third one falls out for free.

Double-angle formulae

sin⁡2θ=2sin⁡θcos⁡θ\sin 2\theta = 2\sin\theta\cos\theta cos⁡2θ=cos⁡2θ−sin⁡2θ=1−2sin⁡2θ\cos 2\theta = \cos^2\theta - \sin^2\theta = 1 - 2\sin^2\theta

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

When to use which

If you see…Reach for…
sin⁡2θ+cos⁡2θ\sin^2\theta + \cos^2\thetareplace with 11
a lone tan⁡θ\tan\theta in a fractionrewrite as sin⁡θcos⁡θ\frac{\sin\theta}{\cos\theta}
sin⁡2θ\sin^2\theta inside an integralthe cos⁡2θ\cos 2\theta identity
sin⁡θcos⁡θ\sin\theta\cos\thetahalf of sin⁡2θ\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.

Free revision pack

Want every formula sheet in one place?

Get this note and the full set as a printable revision pack — plus a new idea each week. Free.