<s>[INST] Prove that $\cos 2x + \cos 2y + \cos 2z = -4(\cos x \cos y \cos z) -1$ when $x+y+z=\pi$ I found on List of trigonometric identities on wikipedia that $\cos 2x + \cos 2y + \cos 2z = -4(\cos x \cos y \cos z) -1$ if $x+y+z=\pi = 180°$. I couldn't find a proof.
 [/INST] We have $\cos2z=\cos2(\pi-x-y)=\cos2(x+y)=2\cos^2(x+y)-1$. Then use the sum-to-product identity twice:
$$\begin{align}
\cos2x+\cos2y+\cos2z&=2\cos(x+y)\cos(x-y)+2\cos^2(x+y)-1 \\
&=2\cos(x+y)\bigl(\cos(x-y)+\cos(x+y)\bigr)-1 \\
&=4\cos(x+y)\cos x\cos y-1 \\
&=4\cos(\pi-z)\cos x\cos y-1 \\
&=-4\cos z\cos x\cos y-1 \\
&=-4\cos x\cos y\cos z-1
\end{align}$$
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