# Find the general solution of each of the following equations:

Question:

Find the general solution of each of the following equations:

cot x + tan x = 2 cosec x

Solution:

To Find: General solution.

Given: $\cot x+\tan x=2 \operatorname{cosec} x \Rightarrow \cos ^{2} x+\sin ^{2} x=2 \sin x \cos x \operatorname{cosec} x \Rightarrow 1=\sin 2 x$ cosec $x$

$\Rightarrow \operatorname{cosec} 2 x=\operatorname{cosec} x \Rightarrow \sin x=\sin 2 x \Rightarrow \sin x=2 \sin x \cos x \Rightarrow \sin x=0$ or $\cos$

$x=\frac{1}{2}=\cos (1)$

Formula used: $\sin \theta=0 \Rightarrow \theta=n \pi, \cos \theta=\cos \alpha \Rightarrow \theta=2 n$

By using above formula, we have

$x=n \pi$ or $x=2 m \pi \pm \frac{\pi}{3}$ where $n, m$ I

So general solution is $x=n \pi$ or $x=2 m \pi \pm \frac{\pi}{3}$ where $n, m \in$ ।