Let S = { x : x 0 , 2 } . The sum of all distinct solutions of the equation 3 sec x + c o s e c x + 2 ( tan x - cot x ) = 0 in the set S is equal to [2016]

Question

Let $S = {x \in (- π, π) : x \neq 0, \pm \frac{π}{2}}$ . The sum of all distinct solutions of the equation $\sqrt{3} \sec x + c o s e c x + 2 (\tan x - \cot x) = 0$ in the set $S$ is equal to [2016]

Smart Achievers · Accepted Answer

(3)

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

$\Rightarrow \frac{\sqrt{3}}{2} \sin x + \frac{1}{2} \cos x = \cos^{2} x - \sin^{2} x$

$\Rightarrow \cos (x - \frac{π}{3}) = \cos 2 x$ $\Rightarrow x - \frac{π}{3} = 2 n π \pm 2 x$

$\Rightarrow x = \frac{2 n π}{3} + \frac{π}{9} or x = - 2 n π - \frac{π}{3}$

$For x \in S, n = 0 \Rightarrow x = \frac{π}{9}, - \frac{π}{3}$

$Now, n = 1 \Rightarrow x = \frac{7 π}{9} and n = - 1 \Rightarrow x = - \frac{5 π}{9}$

$Hence, sum of all values of x = \frac{π}{9} - \frac{π}{3} + \frac{7 π}{9} - \frac{5 π}{9} = 0$

Solution

Q.
Let $S = {x \in (- π, π) : x \neq 0, \pm \frac{π}{2}}$ . The sum of all distinct solutions of the equation $\sqrt{3} \sec x + c o s e c x + 2 (\tan x - \cot x) = 0$ in the set $S$ is equal to [2016]

1 $- \frac{7 π}{9}$

2 $- \frac{2 π}{9}$

3 $0$

4 $\frac{5 π}{9}$

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Solution

Q. Let S={x∈(-π,π):x≠0,±π2}. The sum of all distinct solutions of the equation 3secx+cosec x+2(tanx-cotx)=0 in the set S is equal to [2016]

1 -7π9

2 -2π9

3 0

4 5π9