If [S = x [ 0,2 ]:| pmatrix 0 & x & - x x & 0 & x x & x & 0 pmatrix | = 0 ,] then [_x S ( …

Question

If

$$S = \left\{ {x \in \left[ {0,2\pi } \right]:\left| {\begin{pmatrix} 0 & {\cos x} & { - \sin x}  \
   {\sin x} & 0 & {\cos x}  \
   {\cos x} & {\sin x} & 0  \ \end{pmatrix} } \right| = 0} \right\},$$

then $$\sum\limits_{x \in S} {\tan \left( {{\pi  \over 3} + x} \right)} $$ is equal to :

Accepted Answer

Given,

$$\left| {\begin{pmatrix} 0 & {\cos x} & { - \sin x}  \
   {\sin x} & 0 & {\cos x}  \
   {\cos x} & {\sin x} & 0  \

\end{pmatrix} } \right|$$ = 0

$$ \Rightarrow $$ $\,\,\,$0 (0$-$cosx sinx)$-$cosx (0$-$cos2x)$-$sinx(sin2x) = 0$ \Rightarrow $$ $\,\,\,$$ cos3x - sin3 x = 0

$$ \Rightarrow $$ $\,\,\,$tan3x = 1$ \Rightarrow $$ $\,\,\,$$ tanx = 1

$$ \therefore $$ $$\,\,\,$$  $$\sum\limits_{x\, \in \,\,S} {\,\tan \left( {{\pi  \over 3} + x} \right)} $$

= $${{\tan {\pi  \over 3} + \tan x} \over {1 - \tan {\pi  \over 3}\tan x}}$$

= $${{\sqrt 3  + 1} \over {1 - \sqrt 3 }}$$

= $${{\left( {\sqrt 3  + 1} \right)} \over {\left( {1 - \sqrt 3 } \right)}} \times {{1 + \sqrt 3 } \over {1 + \sqrt 3 }}$$

= $${{1 + 3 + 2\sqrt 3 } \over { - 2}}$$

= - 2 - $$\sqrt 3 $$

Solutions · 1