Question

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

Answer 1

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

⇒ \(\frac{\sqrt 3}2 sin x + \frac 12 cos x + (sin^2x - cos^2x)=0\)

⇒ \(cos (x - \frac \pi3) - cos2x = 0\)

⇒ \(cos (x - \frac \pi3) = cos2x, x = (-\pi,\pi), x \ne 0 , \pm \frac \pi 2\)

⇒ \(x - \frac \pi3 = 2x \; or\; x - \frac \pi3 = -2x \;or\) 

\(x-\frac \pi3 = 2\pi - 2x \;or\;x - \frac \pi3 = - 2\pi + 2x\)

⇒ \(x= -\frac \pi3 \; or\;x = \frac \pi 9\;or\;x = \frac{7 \pi}9 \;or\; x = \frac {9\pi}3\)

sum = \(- \frac \pi3 + \frac {5\pi}3 + \frac \pi 9 + \frac {7\pi}9\)

\(= \frac {4\pi}3 + \frac {8 \pi}9 = \frac {20 \pi}9\)