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7.6PureStretch

Proving Trigonometric Identities

Proving identities combines all the formulae you have met: Pythagorean, addition, double-angle. The method is to start from one side and manipulate it until it matches the other.

26 min Trigonometry and Modelling
Edexcel A level Maths: 7.6 Proving 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

  • Work from one side to the other
  • Choose the right identity at each step
  • Combine fractions and use sin²+cos²=1
  • Lay out a clear, logical proof
1

Method

Start with the more complicated side. Convert to sin and cos where helpful, combine over a common denominator, and apply known identities until it equals the other side.

LHS→⋯→RHS\text{LHS} \to \cdots \to \text{RHS}
Transform one side into the other.
Example 1Prove sin⁡2θ1+cos⁡2θ=tan⁡θ\dfrac{\sin 2\theta}{1 + \cos 2\theta} = \tan\theta.
1Use sin⁡2θ=2sin⁡θcos⁡θ\sin 2\theta = 2\sin\theta\cos\theta and 1+cos⁡2θ=2cos⁡2θ1 + \cos 2\theta = 2\cos^2\theta.
22sin⁡θcos⁡θ2cos⁡2θ=sin⁡θcos⁡θ=tan⁡θ\dfrac{2\sin\theta\cos\theta}{2\cos^2\theta} = \dfrac{\sin\theta}{\cos\theta} = \tan\theta.
Answer
Identity proved.

Tip — Never work on both sides at once — transform one side only.

Formula recap

sin⁡2θ+cos⁡2θ=1\sin^2\theta + \cos^2\theta = 1
Core identity.
1+cos⁡2θ=2cos⁡2θ1 + \cos 2\theta = 2\cos^2\theta
Power-reduction.
sin⁡2θ=2sin⁡θcos⁡θ\sin 2\theta = 2\sin\theta\cos\theta
Double angle.

Common mistakes to avoid

Manipulating both sides simultaneously.
Start from one side and reach the other.
Cancelling without justification.
Only cancel non-zero common factors; state your steps.

Key takeaways

  • Start from the more complex side.
  • Convert to sin/cos and use known identities.
  • Transform one side into the other — never both at once.

Test yourself

Answer a few questions, then review anything you missed.

Prefer paper? Open the written self-check

Answer each question before opening its answer. This self-check does not record a quiz score.

  1. When proving an identity you should:

    • transform one side to the other
    • change both sides
    • square both sides
    • divide by zero
    Show answer and explanation

    Answer: transform one side to the other

    Standard method.

  2. 1+cos⁡2θ1 + \cos 2\theta equals:

    • 2cos⁡2θ2\cos^2\theta
    • 2sin⁡2θ2\sin^2\theta
    • cos⁡2θ\cos^2\theta
    • 2cos⁡θ2\cos\theta
    Show answer and explanation

    Answer: 2cos⁡2θ2\cos^2\theta

    From 2cos²−1.

  3. sin⁡2θ1+cos⁡2θ\dfrac{\sin 2\theta}{1 + \cos 2\theta} simplifies to:

    • tan⁡θ\tan\theta
    • cot⁡θ\cot\theta
    • sin⁡θ\sin\theta
    • 2tan⁡θ2\tan\theta
    Show answer and explanation

    Answer: tan⁡θ\tan\theta

    Cancels to sinθ/cosθ.

  4. A good first step is usually to:

    • start from the complex side
    • guess the answer
    • use degrees
    • integrate
    Show answer and explanation

    Answer: start from the complex side

    More room to simplify.

  5. The most-used identity in proofs is:

    • sin⁡2θ+cos⁡2θ=1\sin^2\theta + \cos^2\theta = 1
    • the sine rule
    • the binomial theorem
    • log laws
    Show answer and explanation

    Answer: sin⁡2θ+cos⁡2θ=1\sin^2\theta + \cos^2\theta = 1

    Pythagorean identity.

Next step

Check your understanding

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

1 past-paper question · 3 marks

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