Question

Prove Cos3x=cos^3(x)-3sin^2(x)cos(x)

Answer 1

$\cos3x=\cos^3x-3\sin^2x\cos x\\\\L_s=\cos(2x+x)=\cos2x\cos x-\sin2x\sin x\\\\=(\cos^2x-\sin^2x)\cos x-2\sin x\cos x\sin x\\\\=\cos^3x-\sin^2x\cos x-2\sin^2x\cos x\\\\=\cos^3x-3\sin^2x\cos x=R_s\\\\\text{Used:}\\\\\cos(\alpha+\beta)=\cos\alpha\cos\beta-\sin\alpha\sin\beta\\\\\cos2\alpha=\cos^2\alpha-\sin^2\alpha\\\\\sin2\alpha=2\sin\alpha\cos\alpha$