Identity proofs and permitted angles

Start from one side of a proposed identity and transform it using established identities and valid algebra. Track any division: a denominator must be nonzero. The familiar identity sin²θ + cos²θ = 1 helps factor differences of squares; tan θ = sin θ/cos θ is defined only when cos θ is nonzero.

Cancelling a factor does not restore angles excluded by the original expression. A simplified formula can have a larger domain than the original one. Check the initial expression and any intermediate denominator separately. A numerical check can disprove an identity by finding a counterexample. A proof must cover every permitted angle.

Worked example

Prove (1 − cos²θ)/(sin θ cos θ) = tan θ and state where the original expression is defined.

1. The original denominator requires sin θ ≠ 0 and cos θ ≠ 0.
2. Replace 1 − cos²θ by sin²θ, giving sin²θ/(sin θ cos θ).
3. Cancel sin θ on the permitted domain to obtain sin θ/cos θ = tan θ.

Answer: The identity holds where sin θ and cos θ are both nonzero; in degrees, exclude all multiples of 90°.

Revise first: Trigonometric identities, Algebraic fractions.