Trigonometrical Equations 1 Question 11
11. Let $S=x \in(-\pi, \pi): x \neq 0, \pm \frac{\pi}{2}$. 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
(2016 Adv.)
(a) $-\frac{7 \pi}{9}$
(b) $-\frac{2 \pi}{9}$
(c) 0
(d) $\frac{5 \pi}{9}$
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Answer:
Correct Answer: 11. (d)
Solution:
- Given, $\sqrt{3} \sec x+\operatorname{cosec} x+2(\tan x-\cot x)=0$,
$(-\pi<x<\pi)-{0, \pm \pi / 2}$
$\Rightarrow \quad \sqrt{3} \sin x+\cos x+2\left(\sin ^{2} x-\cos ^{2} x\right)=0$
$\Rightarrow \quad \sqrt{3} \sin x+\cos x-2 \cos 2 x=0$
Multiplying and dividing by $\sqrt{a^{2}+b^{2}}$, i.e. $\sqrt{3+1}=2$, we get
$$ 2 \frac{\sqrt{3}}{2} \sin x+\frac{1}{2} \cos x \quad-2 \cos 2 x=0 $$
$\Rightarrow \quad \cos x \cdot \cos \frac{\pi}{3}+\sin x \cdot \sin \frac{\pi}{3} \quad-\cos 2 x=0$
$$ \begin{array}{ll} \Rightarrow & \cos \quad x-\frac{\pi}{3}=\cos 2 x \\ \therefore & 2 x=2 n \pi \pm x-\frac{\pi}{3} \quad \stackrel{\text { since, } \cos \theta=\cos \alpha}{\Rightarrow}=2 n \pi \pm \alpha \\ \Rightarrow & 2 x=2 n \pi+x-\frac{\pi}{3} \\ \text { or } & x=2 n \pi-x+\frac{\pi}{3} \\ \Rightarrow & x=2 n \pi-\frac{\pi}{3} \text { or } 3 x=2 n \pi+\frac{\pi}{3} \\ \Rightarrow & x=\frac{2 n \pi}{3}+\frac{\pi}{9} \end{array} $$
$$ \therefore \quad x=\frac{-\pi}{3} \text { or } x=\frac{\pi}{9}, \frac{-5 \pi}{9}, \frac{7 \pi}{9} $$
Now, sum of all distinct solutions
$$ =\frac{-\pi}{3}+\frac{\pi}{9}-\frac{5 \pi}{9}+\frac{7 \pi}{9}=0 $$
