<s>[INST] 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
 [/INST] There were two corrections in the posting ($1102$ should be $1120$ and $\cos x$ term should be $\cos^2 x$ term:
$$\cos 10x = 512 (\cos x)^{10} -1280 (\cos x)^8 +1120 (\cos x)^6 -400(\cos x)^4+50(\cos x)^2-1$$
Follow Arturo's answer you should get the following
$$
\begin{align*}
\cos 2x &= 2 \cos^2 x -1 \\
\cos 3x &= 4 \cos^3 x - 3 \cos x\\
\cos 4x &= 8 \cos^4 x - 8 \cos^2 x +1 \\
\cos 5x &= 16 \cos^5 x - 20 \cos^3 x + 5 \cos x\\
\cos 6x &= 32 \cos^6 x - 48 \cos^4 x + 18 \cos^2 x -1\\
\cos 8x &= 128 \cos^8 x - 256 \cos^6 x + 160 \cos^4 x -32 \cos^2 x +1\\
\cos 10x &= 512 \cos^{10} x -1280 \cos^8 x +1120 \cos^6 x -400 \cos^4 x +50 \cos^2 x-1\\
\end{align*}
$$
(Corrected $-1120 $ to $1120$ )
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