Question

If $S = \left\{x \in \left[0, 2 \pi\right] : \begin{vmatrix}0&\cos x& - \sin x\\ \sin x&0&\cos x\\ \cos x&\sin x&0\end{vmatrix} = 0\right\}$ then $ \sum_{x \epsilon S} \tan\left(\frac{\pi}{3} + x\right) $ is equal to :

• A. $ 4 + 2 \sqrt{3}$
• B. $ - 2 + \sqrt{3}$
• C. $ - 2 - \sqrt{3}$
• D. $ - 4 - 2 \sqrt{3}$

Answer 1

Answer: (C)

$0\left(0 - cosx\right) - cosx\left(0 - cos^{2}x\right) - sinx \left(sin^{2}x - 0\right) = 0$
$cos^{3}x - sin^{3}x = 0$
$tan^{3} = 1 \Rightarrow tanx = 1$
$\sum\frac{\sqrt{3}+tan\,x}{1-\sqrt{3}}$
$\sum\frac{\sqrt{3}+1}{1-\sqrt{3}}\times\frac{1+\sqrt{3}}{1+\sqrt{3}} \Rightarrow \sum\frac{1+3+2\sqrt{3}}{-2} = \sum\frac{4^{2}}{-2}-\frac{2\sqrt{3}}{3}$
$\sum-2-\sqrt{3}$