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 Video by Zeeshan Zamurred Trigonometric Identities and Equations

Edexcel Trig Identities & Equations — full chapter playlist (Zeeshan Zamurred)Watch the full walkthrough 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

Identity-first equations

When an equation contains both $sin$ and $cos$ (or a $tan$ alongside them), use $tan θ = \frac{s i n θ}{c o s θ}$ or $sin^{2} θ + cos^{2} θ = 1$ to express everything in one ratio. Then it becomes a standard equation.

1Divide both sides by

cos θ

tan θ = 1

2Solve:

θ = 4 5^{\circ}, 22 5^{\circ}

Answer

θ = 4 5^{\circ}, 22 5^{\circ}

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

Equations using the Pythagorean identity

If an equation has $sin^{2} θ$ and $cos θ$ together, swap $sin^{2} θ$ for $1 - cos^{2} θ$ to form a quadratic in $cos θ$ , then solve as usual.

1Replace

cos^{2} θ

with

1 - sin^{2} θ

2 (1 - sin^{2} θ) + sin θ = 1

2Tidy to a quadratic:

2 sin^{2} θ - sin θ - 1 = 0

Answerquadratic

2 sin^{2} θ - sin θ - 1 = 0

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 \Rightarrow tan θ = 1

Divide by cos to simplify.

sin^{2} θ + cos^{2} θ \equiv 1

Form a quadratic in one ratio.

tan θ \equiv \frac{sin θ}{cos θ}

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

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