Proving Trigonometric Identities

This lesson focuses on strategies and techniques for proving trigonometric identities. Identity proofs appear frequently in A-Level examinations, and mastering them requires both a solid knowledge of standard identities and systematic problem-solving strategies.

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What Is a Trigonometric Identity?

A trigonometric identity is an equation that is true for all values of the variable in its domain. We write identities using the symbol ≡ (identically equal to), though the = sign is also accepted in examinations.

For example, sin²θ + cos²θ ≡ 1 is true for every real number θ.

An identity is different from an equation, which is only true for specific values. For example, sin θ = 1/2 is only true for certain values of θ.

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The Toolkit: Identities You Must Know

Fundamental Identities

sin²θ + cos²θ ≡ 1
tan θ ≡ sin θ / cos θ

Pythagorean Identities

1 + tan²θ ≡ sec²θ
1 + cot²θ ≡ cosec²θ

Double Angle Formulae

sin 2A ≡ 2sin A cos A
cos 2A ≡ cos²A − sin²A ≡ 2cos²A − 1 ≡ 1 − 2sin²A
tan 2A ≡ 2tan A / (1 − tan²A)