Cos and sin

$\cos(x) + \cos(y) + \cos(z) + i(\sin(x) + \sin(y) + \sin(z)) = 0$
$e^{ix} + e^{iy} + e^{iz} = 0$
$\cos(x) + \cos(y) + \cos(z) - i(\sin(x) + \sin(y) + \sin(z)) = 0$
$e^{-ix} + e^{-iy} + e^{-iz} = 0$
$e^{i(x+y)} + e^{i(x+z)} + e^{i(z+y)} = 0$
$(e^{ix} + e^{iy} + e^{iz})^{2} = e^{i2x} + e^{i2y} + e ^{i2z} + 2(e^{i(x+y)} + e^{i(x+z)} + e^{i(z+y)})$
$\therefore e^{i2x} + e^{i2y} + e ^{i2z} = 0$
$\cos(2x) + \cos(2y) + \cos(2z) + i(\sin(2x) + \sin(2y) + \sin(2z)) = 0$
$\cos(2x) + \cos(2y) + \cos(2z) = 0$
$\sin(2x) + \sin(2y) + \sin(2z) = 0$
Infact,
$\cos(2^{n}x) + \cos(2^{n}y) + \cos(2^{n}z) = 0$
$\sin(2^{n}x) + \sin(2^{n}y) + \sin(2^{n}z) = 0$ $n \in Z^{+}$