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 Video by Zeeshan Zamurred Trigonometry and Modelling

Edexcel A level Maths: 7.6 Proving Trigonometric IdentitiesWatch the full walkthrough before the notes below.

Open on YouTube

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

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 \to \dots \to RHS

Transform one side into the other.

1Use

sin 2 θ = 2 sin θ cos θ

and

1 + cos 2 θ = 2 cos^{2} θ

\frac{2 sin θ cos θ}{2 cos ^{2} θ} = \frac{sin θ}{cos θ} = tan θ

AnswerIdentity proved.

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

Formula recap

sin^{2} θ + cos^{2} θ = 1

Core identity.

1 + cos 2 θ = 2 cos^{2} θ

Power-reduction.

sin 2 θ = 2 sin θ cos θ

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

Ready to lock in Proving Trigonometric Identities? Pick a mode and earn XP & Dobloons.