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TrigonometryTrigonometryHigh school

Trigonometric Identities

Pythagorean, double-angle and sum identities.

A trigonometric identity is an equation that holds for every angle where both sides are defined. The Pythagorean identity sin⁡2θ+cos⁡2θ=1\sin^2\theta + \cos^2\theta = 1 comes straight from the unit circle, and dividing it by cos⁡2θ\cos^2\theta or sin⁡2θ\sin^2\theta gives two more: 1+tan⁡2θ=sec⁡2θ1 + \tan^2\theta = \sec^2\theta and 1+cot⁡2θ=csc⁡2θ1 + \cot^2\theta = \csc^2\theta.

Sum and double-angle formulas such as sin⁡2θ=2sin⁡θcos⁡θ\sin 2\theta = 2\sin\theta\cos\theta turn awkward expressions into simpler ones, and they are essential later for integrals. To prove an identity, work on one side only, usually the more complicated one, rewriting in sines and cosines until it matches the other.

Key terms

Pythagorean identity
Holds for every angle.
sin⁡2θ+cos⁡2θ=1\sin^2\theta + \cos^2\theta = 1

Key ideas

  • Rewrite everything in sine and cosine when stuck.
  • cos⁡2θ\cos 2\theta has three forms: cos⁡2θ−sin⁡2θ\cos^2\theta - \sin^2\theta, 2cos⁡2θ−12\cos^2\theta - 1, 1−2sin⁡2θ1 - 2\sin^2\theta.
  • Factor and use common denominators exactly as in algebra.
  • Never move terms across the equals sign when proving; transform one side.

Worked example

Prove 1−cos⁡2xsin⁡xcos⁡x=tan⁡x\frac{1 - \cos^2 x}{\sin x \cos x} = \tan x.

  1. Numerator: 1−cos⁡2x=sin⁡2x1 - \cos^2 x = \sin^2 x.
  2. So the left side is sin⁡2xsin⁡xcos⁡x\frac{\sin^2 x}{\sin x \cos x}.
  3. Cancel one sin⁡x\sin x: sin⁡xcos⁡x\frac{\sin x}{\cos x}.
  4. That equals tan⁡x\tan x.

Answer: Identity holds wherever sin⁡xcos⁡x≠0\sin x \cos x \ne 0

Common mistakes

  • Writing sin⁡(a+b)=sin⁡a+sin⁡b\sin(a + b) = \sin a + \sin b.
  • Treating sin⁡2x\sin^2 x as sin⁡(x2)\sin(x^2).
  • Proving by assuming the identity and manipulating both sides together.

Practice questions

Try each one, then open the answer. Numerical answers on this page are verified by an automated calculation check.

  1. 1.If sin⁡θ=35\sin\theta = \frac{3}{5} with θ\theta acute, find cos⁡θ\cos\theta.

    Show answer

    45\frac{4}{5}

  2. 2.Evaluate 2sin⁡15∘cos⁡15∘2\sin 15^\circ \cos 15^\circ.

    Show answer

    12\frac{1}{2}

  3. 3.Evaluate cos⁡222.5∘−sin⁡222.5∘\cos^2 22.5^\circ - \sin^2 22.5^\circ to three decimals.

    Show answer

    22≈0.707\frac{\sqrt{2}}{2} \approx 0.707

Keep practising

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