In order to show that $ e^{ix}+e^{iy}+e^{iz}=0 \Longrightarrow e^{2ix}+e^{2iy}+e^{2iz}=0 $, I want to prove that $ \cos x+\cos y+\cos z=0 $ and $ \sin x+\sin y+\sin z=0 \Longrightarrow \cos 2x+\cos 2y+\cos 2z=0$ and $ \sin 2x+\sin 2y+\sin 2z=0 $
$ \cos 2x=2\cos^2 x-1=2(\cos y+\cos z)^2-1 $
$ \cos 2x+\cos 2y+\cos 2z=2(\cos^2x+\cos^2y+\cos^2z+$
$(\cos^2x+cos^2y+\cos^2z+2(\cos x\cos y+\cos y\cos z+\cos x\cos z)))-3 $
$ \cos 2x+\cos 2y+\cos 2z=2(\cos^2x+\cos^2y+\cos^2z)-3 =3-2(\sin^2x+\sin^2y+\sin^2z)$ ...
Any idea?