Express $\sin4\theta$ in terms of powers of $\sin\theta$ and $\cos\theta$

Question

As far as I know $\sin4\theta$ = $4\sin\theta \cos\theta$, but I don't know if that's correct or what to do from there?

Answer 1

$$ \sin 4\theta = \sin (2\times 2\theta) = 2 \sin 2\theta \cos 2\theta=2(2\sin\theta \cos \theta)(\cos^2\theta- \sin^2 \theta) \\= 4 \sin \theta \cos \theta (\cos ^2 \theta - \sin ^2 \theta) $$

Answer 2

Hints:

$\sin 2\theta=2\sin \theta \cos \theta.$

$\cos 2\theta=\cos^2\theta-\sin^2\theta=2\cos^2\theta-1=1-2\sin^2\theta$