How to expand $\cos nx$ with $\cos x$? Multiple Angle Identities:
How to expand  $\cos nx$  with  $\cos x$, such as 
$$\cos10x=512(\cos x)^{10}-1280(\cos x)^8+1120(\cos x)^6-400(\cos x)^4+50(\cos x)^2-1$$
See a list of trigonometric identities in english/ chinese

You can always repeatedly use
$$\begin{align*}
\cos(a\pm b) &= \cos a\cos b \mp \sin a\sin b\\
\sin(a\pm b) &= \sin a\cos b \pm \cos a\sin b\\
\sin^2(r) &= 1-\cos^2(r).
\end{align*}$$
For example,
$$\begin{align*}
\cos(4x) &= \cos(2x+2x)\\
&= \cos(2x)^2 - \sin^2(2x)\\
&= \cos(2x)^2 - (1-\cos^2(2x))\\
&= 2\cos(2x)^2 - 1\\
&= 2(\cos(x+x))^2 - 1\\
&= 2(\cos x\cos x - \sin x\sin x)^2 - 1\\
&= 2(\cos^2 x - \sin^2 x)^2 - 1\\
&= 2(\cos^2x - (1-\cos^2 x))^2 - 1\\
&= 2(2\cos^2x - 1)^2 - 1\\
&= 2(4\cos^4 x - 4\cos^2 x + 1) - 1\\
&= 8\cos^4 x - 8\cos^2 x + 1.
\end{align*}$$
