Sometimes, its better to adopt a complex way than a simpler one

Geometry Level 5

$\sin(x)+\sin(y)+\sin(z)=0 \\ \cos(x)+\cos(y)+\cos(z)=0$

Given that $x,y,z$ are real numbers satisfying above two equations , evaluate:

$\begin{aligned} &\tan^2(x)+\tan^2(y)+\tan^2(z) \\ - &\tan^2(x)\tan^2(y)-\tan^2(y)\tan^2(z)-\tan^2(x)\tan^2(z) \\ -&3\tan^2(x)\tan^2(y)\tan^2(z) = \ ? \end{aligned}$

Sometimes, its better to adopt a complex way than a simpler one

This problem is original.