Question
Let $$S = \left\{ {x \in \left( { - \pi ,\pi } \right):x \ne 0, \pm \frac{\pi }{2}} \right\}.$$ The sum of all distinct solutions of the equation $$\sqrt 3 \sec x + {\text{cosec}}\,x + 2\left( {\tan x - \cot x} \right) = 0$$ in the set $$S$$ is equal to
A.
$$ - \frac{{7\pi }}{9}$$
B.
$$ - \frac{{2\pi }}{9}$$
C.
0
D.
$$ \frac{{5\pi }}{9}$$
Answer :
0
Solution :
$$\eqalign{
& \sqrt 3 \,\sec \,x\, + \,{\text{cosec}}\,x\, + 2\,\left( {\tan \,x\, - \cot \,x} \right) = 0 \cr
& \Rightarrow \,\,\frac{{\sqrt 3 }}{2}\,\sin \,x\, + \frac{1}{2}\cos \,x\,\, = {\cos ^2}x - \,{\sin ^2}x \cr
& \Rightarrow \,\,\cos \,\left( {x\, - \frac{\pi }{3}} \right)\, = \cos \,2x \cr
& \Rightarrow \,\,x - \frac{\pi }{3}\, = 2n\pi \pm 2x \cr
& \Rightarrow \,\,x = \frac{{2n\pi }}{3} + \frac{\pi }{9}\,\,{\text{or}}\,\,x = \, - 2n\pi \, - \frac{\pi }{3} \cr
& {\text{For}}\,x\, \in \,S{\text{,}}\,n = {\text{0}} \cr
& \Rightarrow \,\,x = \frac{\pi }{9},\, - \frac{\pi }{3} \cr
& n = 1 \cr
& \Rightarrow \,\,x = \frac{{7\pi }}{9};\, \cr
& n = - 1 \cr
& \Rightarrow \,\,x = \frac{{ - 5\pi }}{9} \cr} $$
∴ Sum of all values of $$x = \frac{\pi }{9} - \frac{\pi }{3} + \frac{{7\pi }}{9} - \frac{{5\pi }}{9} = 0$$