<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] $$
\text{LHS}{=\cos 2x+\cos 2y+\cos (2x+2y)
\\=\cos 2x+\cos 2y+\cos 2x\cos 2y-\sin 2x\sin 2y
}
$$
$$
\text{RHS}{=4\cos x\cos y\cos(x+y)-1
\\=4\cos^2 x\cos^2 y-4\cos x\cos y\sin x\sin y-1
\\=(\cos 2x+1)(\cos 2y+1)-\sin 2x\sin 2y-1
\\=\cos 2x+\cos 2y+\cos 2x\cos 2y-\sin 2x\sin 2y
}
$$
which completes the proof $\blacksquare$
</s>