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જો $S = \left\{ {x \in \left[ {0,2\pi } \right]:\left| {\begin{array}{*{20}{c}}
0&{\cos {\mkern 1mu} x}&{ - \sin {\mkern 1mu} x}\\
{\sin {\mkern 1mu} x}&0&{\cos {\mkern 1mu} x}\\
{\cos {\mkern 1mu} x}&{\sin {\mkern 1mu} x}&0
\end{array}} \right| = 0} \right\},$ તો $\sum\limits_{x \in S} {\tan \left( {\frac{\pi }{3} + x} \right)} $ =
$4 + 2\sqrt 3 $
$-2 + \sqrt 3 $
$-2 - \sqrt 3 $
$-4 - 2\sqrt 3 $
Solution
since the given determinant is equal to zera
$\Rightarrow 0(0-\cos x \sin x)-\cos x\left(0-\cos ^{2} x\right)$
$-\sin x\left(\sin ^{2} x-0\right)=0$
$\Rightarrow \cos ^{3} x-\sin ^{3} x=0$
$\Rightarrow \tan ^{3}=1 \Rightarrow \tan x=1$
$\therefore \quad \sum_{x \in s} \tan \left(\frac{\pi}{3}+x\right)=\sum_{x \in s} \frac{\tan \pi / 3+\tan x}{1-\tan \pi / 3 \cdot \tan x}$
${ = \sum\limits_{x\, \in \,s} {\frac{{\sqrt 3 \, + \,1}}{{1\, – \,\sqrt 3 }}\, = \sum\limits_{x\, \in \,s} {\frac{{\sqrt 3 \, + \,1}}{{1\, – \,\sqrt 3 }}\, \times \,\frac{{1 + \sqrt 3 }}{{1 + \sqrt 3 }}} \,} }$
${ \Rightarrow \sum\limits_{x \in s} {\frac{{1 + 3 + 2\sqrt 3 }}{{ – 2}}} = – 2 – \sqrt 3 }$
