Trigonometric proofs using De Moivre’s theorem

Prove multi-angle trigonometric identities up to angles of 4θ by equating parts using the binomial expansion and De Moivre’s theorem, e.g. cos(3θ) = 4 cos³(θ) − 3 cos(θ) and sin(3θ) = 3 sin(θ) − 4 sin³(θ).