Find the general solution of each of the following equations:

To Find: General solution.

Given: $2 \tan x-\cot x+1=0 \Rightarrow 2 \tan ^{2} x-1+\tan x=0 \Rightarrow 2 \tan ^{2} x-1+2 \tan x-\tan x=$ $0 \Rightarrow 2 \tan x(\tan x+1)-(1+\tan x)=0$

$\Rightarrow(2 \tan x-1)(1+\tan x)=0 \Rightarrow \tan x=\frac{1}{2}=\tan ^{-1} \frac{1}{2}$ or $\tan x=-1=\tan \frac{3 \pi}{4}$

Formula used: $\tan \theta=\tan \alpha \Rightarrow \theta=\mathrm{n} \pi+\alpha, \mathrm{n} \in I$

$x=n \pi+\tan ^{-1} \frac{1}{2}$ or $x=n \pi+\frac{3 \pi}{4}$

So the general solution is $x=n \pi+\tan ^{-1} \frac{1}{2}$ or $x=n \pi+\frac{3 \pi}{4}$ where $n \in I$