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 Video by Zeeshan Zamurred Trigonometric Identities and Equations

Edexcel AS Level Maths: 10.3 Trigonometric IdentitiesWatch the full walkthrough 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

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 $tan$ .

sin^{2} θ + cos^{2} θ \equiv 1

The Pythagorean identity — true for all θ.

tan θ \equiv \frac{sin θ}{cos θ}

The tangent identity.

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

Simplifying expressions

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

\frac{sin θ}{cos θ} = tan θ

, but here the

cos θ

cancels.

2Left with

sin θ

Answer

sin θ

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 $\equiv$ .

1From

sin^{2} θ + cos^{2} θ \equiv 1

, subtract

cos^{2} θ

2This gives

sin^{2} θ = 1 - cos^{2} θ

Answer

1 - cos^{2} θ = sin^{2} θ

∎

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

Formula recap

sin^{2} θ + cos^{2} θ \equiv 1

Pythagorean identity.

tan θ \equiv \frac{sin θ}{cos θ}

Tangent identity.

sin^{2} θ = 1 - cos^{2} θ

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.

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