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

Geometry Level 5

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

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

tan2(x)+tan2(y)+tan2(z)tan2(x)tan2(y)tan2(y)tan2(z)tan2(x)tan2(z)3tan2(x)tan2(y)tan2(z)= ?\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}


This problem is original.
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