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Prove it: trig identities

Proving is choosing the right next move. Build each proof line by line — and learn the five moves that crack almost every identity.
The five moves
  1. Turn everything into sin and cos.
  2. Spot Pythagoras in disguise (1 − cos²θ, 1 − sin²θ, sec²θ − 1…)
  3. Combine fractions over one denominator.
  4. Multiply by the conjugate (1 ± sin θ, 1 ± cos θ)
  5. Factorise and cancel — legally.
Identities I may use