Use de Moivre's theorem to express $\sin 6\theta$ in terms of $\sin \theta$ and $\cos \theta$

Question

Use de Moivre's theorem to express $\sin 6\theta$ in terms of $\sin \theta$ and $\cos \theta$ $\cos 6\theta$ in terms of $\cos \theta$.

Accepted Answer

(a) $6\cos^5\theta\sin\theta-20\cos^3\theta\sin^3\theta+6\cos\theta\sin^5\theta$ (b) $32\cos^6\theta-48\cos^4\theta+18\cos^2\theta-1$.

Cosine and sine of multiple angle expressed as polynomials of sines and cosines