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10.3PureStretch

Trigonometric Identities

Two identities underpin almost all of trigonometry at this level: the Pythagorean identity and the tangent identity. They let you rewrite expressions, prove results and turn awkward equations into ones you can solve.

30 min Trigonometric Identities and Equations
Edexcel AS Level Maths: 10.3 Trigonometric IdentitiesCurated by Kepler, not produced by us. Watch it before the notes below.
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What you'll be able to do

  • State and use the Pythagorean identity
  • Use the identity tan θ = sin θ / cos θ
  • Simplify trigonometric expressions
  • Prove trigonometric identities
1

The two key identities

The Pythagorean identity comes straight from the unit circle (it is Pythagoras in disguise). The tangent identity is the definition of tan⁡\tan.

sin⁡2θ+cos⁡2θ≡1\sin^{2}\theta + \cos^{2}\theta \equiv 1
The Pythagorean identity — true for all θ.
tan⁡θ≡sin⁡θcos⁡θ\tan\theta \equiv \dfrac{\sin\theta}{\cos\theta}
The tangent identity.

Tip — Rearranged forms are just as useful: sin²θ = 1 − cos²θ and cos²θ = 1 − sin²θ.

2

Simplifying expressions

Spotting where an identity applies turns a messy expression into something simple. Look especially for sin⁡2+cos⁡2\sin^2 + \cos^2 groupings.

Example 1Simplify sin⁡θcos⁡θ×cos⁡θ\dfrac{\sin\theta}{\cos\theta} \times \cos\theta.
1sin⁡θcos⁡θ=tan⁡θ\dfrac{\sin\theta}{\cos\theta} = \tan\theta, but here the cos⁡θ\cos\theta cancels.
2Left with sin⁡θ\sin\theta.
Answersin⁡θ\sin\theta
3

Proving identities

To prove an identity, start with one side (usually the messier one) and manipulate it using the two identities until it equals the other side. Never move terms across the ≡\equiv.

Example 1Prove (1−cos⁡2θ)(1+tan⁡2θ)(1 - \cos^2\theta)(1 + \tan^2\theta) ... show 1−cos⁡2θ=sin⁡2θ1 - \cos^2\theta = \sin^2\theta.
1From sin⁡2θ+cos⁡2θ≡1\sin^2\theta + \cos^2\theta \equiv 1, subtract cos⁡2θ\cos^2\theta.
2This gives sin⁡2θ=1−cos⁡2θ\sin^2\theta = 1 - \cos^2\theta.
Answer1−cos⁡2θ=sin⁡2θ1 - \cos^2\theta = \sin^2\theta ∎

Tip — When you see 1 − cos²θ or 1 − sin²θ, immediately replace it with sin²θ or cos²θ.

Formula recap

sin⁡2θ+cos⁡2θ≡1\sin^{2}\theta + \cos^{2}\theta \equiv 1
Pythagorean identity.
tan⁡θ≡sin⁡θcos⁡θ\tan\theta \equiv \dfrac{\sin\theta}{\cos\theta}
Tangent identity.
sin⁡2θ=1−cos⁡2θ\sin^{2}\theta = 1 - \cos^{2}\theta
Handy rearrangement.

Common mistakes to avoid

Writing sin²θ as sin θ².
sin²θ means (sin θ)², the whole ratio squared.
Rearranging across the ≡ sign as if solving an equation.
For a proof, transform one side into the other.

Key takeaways

  • sin²θ + cos²θ ≡ 1 and tan θ ≡ sin θ / cos θ are the core identities.
  • Use rearrangements like sin²θ = 1 − cos²θ to simplify.
  • Prove identities by transforming one side into the other.

Test yourself

Answer a few questions, then review anything you missed.

Next step

Check your understanding

Work through questions on Trigonometric Identities. Use the feedback to decide what needs another look.

8 past-paper questions · 19 marks

Practise this topic

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