Trig is fun

$\begin{aligned} \sin x + \sin y + \sin z = \cos x + \cos y + \cos z & = 0 \\ \Rightarrow \cos x + \cos y + \cos z + i(\sin x + \sin y + \sin z) & = 0 \\ e^{ix} + e^{iy} + e^{iz} & = 0\\ \color{#3D99F6}{\text{An trivial solution is the third roots of unity}} \quad \Rightarrow \omega^2 + \omega + 1 & = 0 \\ e^{i\frac{4\pi}{3}} + e^{i\frac{2\pi}{3}} + e^{i0} & = 0 \\ \Rightarrow x, y, z & = 0, \frac{2\pi}{3}, \frac{4 \pi} {3} \\ \Rightarrow \cos^2 x + \cos^2 y + \cos^2 z & = \cos^2 0 + \cos^2 \frac{2\pi}{3} + \cos^2 \frac{4\pi}{3} \\ & = 1^2 + \left(-\frac{1}{2} \right)^2 + \left(-\frac{1}{2} \right)^2 = \boxed{1.5} \end{aligned}$