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10.6PureStretch

Equations and Identities

This final section brings the chapter together: using the two identities to turn tough equations into solvable ones, and writing full, rigorous proofs of trigonometric identities. It is where exam questions live.

30 min Trigonometric Identities and Equations
Edexcel Trig Identities & Equations — full chapter playlist (Zeeshan Zamurred)Curated by Kepler, not produced by us. Watch it before the notes below.
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What you'll be able to do

  • Use identities to reduce an equation to one ratio
  • Solve equations that need an identity first
  • Prove identities cleanly and fully
  • Choose the most efficient route through a problem
1

Identity-first equations

When an equation contains both sin⁡\sin and cos⁡\cos (or a tan⁡\tan alongside them), use tan⁡θ=sin⁡θcos⁡θ\tan\theta = \tfrac{\sin\theta}{\cos\theta} or sin⁡2θ+cos⁡2θ=1\sin^2\theta + \cos^2\theta = 1 to express everything in one ratio. Then it becomes a standard equation.

Example 1Solve sin⁡θ=cos⁡θ\sin\theta = \cos\theta for 0∘≤θ≤360∘0^\circ \le \theta \le 360^\circ.
1Divide both sides by cos⁡θ\cos\theta: tan⁡θ=1\tan\theta = 1.
2Solve: θ=45∘,225∘\theta = 45^\circ, 225^\circ.
Answerθ=45∘,225∘\theta = 45^\circ, 225^\circ

Tip — Seeing sinθ = cosθ? Divide by cosθ to get tanθ = 1 — far easier than squaring.

2

Equations using the Pythagorean identity

If an equation has sin⁡2θ\sin^2\theta and cos⁡θ\cos\theta together, swap sin⁡2θ\sin^2\theta for 1−cos⁡2θ1 - \cos^2\theta to form a quadratic in cos⁡θ\cos\theta, then solve as usual.

Example 1Solve 2cos⁡2θ+sin⁡θ=12\cos^2\theta + \sin\theta = 1 ... rewrite in terms of sin⁡θ\sin\theta.
1Replace cos⁡2θ\cos^2\theta with 1−sin⁡2θ1 - \sin^2\theta: 2(1−sin⁡2θ)+sin⁡θ=12(1 - \sin^2\theta) + \sin\theta = 1.
2Tidy to a quadratic: 2sin⁡2θ−sin⁡θ−1=02\sin^2\theta - \sin\theta - 1 = 0.
Answerquadratic 2sin⁡2θ−sin⁡θ−1=02\sin^2\theta - \sin\theta - 1 = 0
3

Writing proofs

For a proof, start from the more complex side and transform it using the identities until it matches the other side. Set work out line by line, and never rearrange across the ≡\equiv sign.

Tip — Convert everything to sin and cos — most identity proofs fall out once you do.

Formula recap

sin⁡θ=cos⁡θ⇒tan⁡θ=1\sin\theta = \cos\theta \Rightarrow \tan\theta = 1
Divide by cos to simplify.
sin⁡2θ+cos⁡2θ≡1\sin^{2}\theta + \cos^{2}\theta \equiv 1
Form a quadratic in one ratio.
tan⁡θ≡sin⁡θcos⁡θ\tan\theta \equiv \dfrac{\sin\theta}{\cos\theta}
Convert tan to sin/cos.

Common mistakes to avoid

Squaring sin θ = cos θ and creating extra (false) solutions.
Divide by cos θ to get tan θ = 1 — no spurious roots.
Leaving an equation mixing sin² and cos unsolved.
Use the identity to get a single ratio, then solve the quadratic.

Key takeaways

  • Use identities to reduce an equation to a single trig ratio.
  • sin θ = cos θ → tan θ = 1.
  • Swap sin²θ for 1 − cos²θ (or vice versa) to form a quadratic.
  • Prove identities by transforming one side, converting to sin and cos.

Test yourself

Answer a few questions, then review anything you missed.

Next step

Keep practising

Start with the quiz above to check the key ideas. You can also choose another topic from the practice library.

Browse practice topics

Related lessons

All 6 lessons

Put the methods into practice: see how A-Level maths duels work. An account is required to play; the overview and sample are public.