Mathematics - Determinant Question with Solution | TestHub

Given,$\left|\begin{array}{ccc}0& \cos x& -\sin x\\ \sin x& 0& \cos x\\ \cos x& \sin x& 0\end{array}\right|=0$

Expanding the determinant,

$\Rightarrow 0(0-\cos x\sin x)-\cos x\left(0-{\cos }^{2}x\right)-\sin x\left({\sin }^{2}x-0\right)=0$

$\Rightarrow$ ${\cos }^{3}x-{\sin }^{3}x=0$

$\Rightarrow$ ${\tan }^{3}x=1\Rightarrow \tan x=1\Rightarrow x=\frac{\pi }{4},\frac{5\pi }{4} \Rightarrow S=\left\{\frac{\pi }{4}, \frac{5\pi }{4}\right\}$

$\therefore \sum _{x\in S}\tan \left(\frac{\pi }{3}+x\right)$

$=\sum _{x\in S}\left(\frac{\sqrt{3}+\tan x}{1-\sqrt{3}\tan x}\right)$

$=2\left(\frac{\sqrt{3}+1}{1-\sqrt{3}}\right)=-4-2\sqrt{3}$