Prove multi-angle trigonometric identities up to angles of 4𝑥 by equating parts using the binomial expansion and De Moivre’s theorem, e.g. cos(3𝑥) = 4 cos3(𝑥) − 3 cos(𝑥) and sin(3𝑥) = 3 sin(𝑥) − 4 sin3(𝑥). Specialist Mathematics 2025 v1.4

Using De Moivre's theorem and the binomial expansion, prove that \(\sin(4x) = 4\cos^3(x)\sin(x) - 4\cos(x)\sin^3(x)\).

Use De Moivre's theorem and the binomial expansion to prove that \(\cos(4x) = 8\cos^4(x) - 8\cos^2(x) + 1\). a) Apply De Moivre's theorem to express \((\cos(x) + i\sin(x))^4\) in the form \(\cos(4x) + i\sin(4x)\). (1 mark) b) Hence prove the identity \(\cos(4x) = 8\cos^4(x) - 8\cos^2(x) + 1\) by equating real parts. (1 mark)

Prove that \(\sin(4x) = 4\sin(x)\cos(x)(1 - 2\sin^2(x))\) using De Moivre's theorem and the binomial theorem.