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Trig identities: why each step

Every line of the proof, with the thing you spotted that made it the right move. Read the plan first, then walk it line by line — the reasoning matters more than the algebra.
Pick a proof to walk through

The five moves — nearly every proof is a sequence of these

  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.
Once a proof makes sense here, go and build one yourself in Prove It: Trig Identities — same identities, but there you choose each line.