Use DeMoivre's Theorem to prove $ \cos 5x = 16 \cos^5x - 20\cos^3x+5\cos x$ I need to prove the following equalities using DMT:
$ \cos 5x  = 16 \cos^5x$$ - 20\cos^3x$$+5\cos x$
and 
$ \sin 5x  = 16 \sin^5x$$ - 20\sin^3x$$+5\sin x$
Can someone help me with this question?
(Attempt: $\cos5x+i\sin5x=(\cos x+i\sin x)^5$)

By DeMoivre's Theorem:

$$\begin{equation}\begin{aligned}
cos\,5x &= cos\,5x + i\,sin\,5x \\
&= (cos\,x + i\,sin\,x)^5 \\
&= cos^5\,x + 10\,cos^3\,sin^2\,x\,i + 5\,cos\,x\,sin^4\,x\,i^4\\
&= cos^5\,x - 10\,cos^3 x\,sin^2\,x + 5\,cos\,x\,sin^4\,x \\
&= cos^5\,x - 10\,cos^3\,x (1 - cos^2\,x) + 5\,cos\,x (1 - cos^2\,x)^2 \\
&= 16\,cos^5\,x - 20\,cos^3\,x + 5\,cos\,x \\
\end{aligned}\end{equation}$$
