<s>[INST] Prove that $16\cos^5A-20\cos^3A+5\cos A=\cos5A$ 
Prove that 
  $$16\cos^5A-20\cos^3A+5\cos A=\cos5A$$

My solution begins here;
$$
\begin{align}
\text{RHS} & =\cos5A \\
& =\cos(A+4A) \\
& =\cos A\cos4A-\sin A\sin4A \\
& =\cos A(2\cos^2 2A-1)-\sin A(2\sin2A\cos2A) \\
& =2\cos A\cos^2 2A-\cos A-2\sin A\sin2A\cos2A \\
& =2\cos A\cos^2 2A-\cos A-2\sin A(2\sin A\cos A)\cos2A
\end{align}
$$
Now how do I move on?
 [/INST] Now use double angle formulae to write $\cos 2A$ in terms of $\cos A$, then turn any $\sin^2 A$ into $1-\cos^2 A$
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