Let `S= {x in (-pi, pi): x ne 0, +- (pi)/(2)}`. The sum of all distinct solutions of the equation `sqrt(3) "sec" x + "cosec"x + 2 ("tan" x - "cot" x) = 0` in the set S is equal to

Options

A. `-(7 pi)/9`
B. `-(2pi)/9`
C. 0
D. `(5pi)/9`

Answer

C

Explanation

`sqrt(3) sec x + cosec x =2 (cot x - tan x)`
`rArr sqrt(3)/(cos x)+1/(sin x) =2 ((cos x)/(sin x)-(sin x)/(cos x))`
`rArr sqrt(3) sin x + cos x =2(cos^(2) x-sin^(2) x)`
`rArr cos (x-pi/3)=cos 2x`
`rArr 2x= 2npi pm (x-pi/3), n in Z`
`rArr x=2n pi - pi/3, (2 n pi)/3 +pi/9`
In `(-pi, pi), x = (-pi)/3, pi/9, (7pi)/9, (-5pi)/9`
`:. Sigmax_(i)=0`