Let $S=\left\{x \in(-\pi, \pi): x \neq 0, \pm \frac{\pi}{2}\right\}$. The sum of all distinct solutions of the equation $\sqrt{3} \sec x+\operatorname{cosec} x+2(\tan x-\cot x)=0$ in the set $S$ is equal to

JEE Advanced · 2016 · 22 May 2016 · Paper 1 · Trigonometric Equations

Let

S = {x \in (- π, π) : x \neq = 0, \pm \frac{π}{2}}

. The sum of all distinct solutions of the equation

3 sec x + cosec x + 2 (tan x - cot x) = 0

in the set

S

is equal to

Answer: (C)

Rewrite the equation:

3 sec x + csc x + 2 (tan x - cot x) = 0

Multiplying through by

sin x cos x

3 sin x + cos x = 2 cos 2 x

This can be written as:

2 (\frac{3}{2} sin x + \frac{1}{2} cos x) = 2 cos 2 x

cos (x - \frac{π}{3}) = cos 2 x

This gives:

x - \frac{π}{3} = 2 nπ \pm 2 x

Case 1:

x - \frac{π}{3} = 2 nπ + 2 x \Rightarrow x = - \frac{π}{3} - 2 nπ

For

x \in (- π, π)

x = - \frac{π}{3}

Case 2:

x - \frac{π}{3} = 2 nπ - 2 x \Rightarrow 3 x = \frac{π}{3} + 2 nπ

For

x \in (- π, π)

x = \frac{π}{9}, - \frac{5 π}{9}, \frac{7 π}{9}

Sum of all solutions:

- \frac{π}{3} + \frac{π}{9} - \frac{5 π}{9} + \frac{7 π}{9} = - \frac{3 π}{9} + \frac{3 π}{9} = 0

JEE Advanced · 2016 · 22 May 2016 · Paper 1 · Single correct option