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:

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)−3tan⁡2(x)tan⁡2(y)tan⁡2(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.
Nihar Mahajan by Nihar Mahajan
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