<s>[INST] 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: $\cos5x+i\sin5x=(\cos x+i\sin x)^5$)
 [/INST] Use the short-hands $c= \cos x$, $s=\sin x$ and continue with 
$$\cos5x+i\sin5x=(c+is)^5
=c^5-10s^2c^3+5s^4c+i(s^5-10s^3c^2+5sc^4)$$
$$=c^5-10(1-c^2)c^3+5(1-c^2)^2c+i(s^5-10s^3(1-s^2)+5s(1-s^2)^2)$$
$$=16c^5-20c^3+5c+i(16s^5-20s^3+5s)$$
where $c^2+s^2=1$ is used. Thus,
$$ \cos 5x  = 16 \cos^5x - 20\cos^3x+5\cos x$$
$$ \sin 5x  = 16 \sin^5x - 20\sin^3x +5\sin x$$
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