Answer
$x=\frac{\pi}{4}+n\pi$
$x=arctan~\frac{1}{2}+n\pi$
where $n$ is an integer.
Work Step by Step
$2~tan^2x-3~tan~x=-1$
$2~tan^2x-2~tan~x-tan~x+1=0$
$2~tan~x(tan~x-1)-(tan~x-1)=0$
$(tan~x-1)(2~tan~x-1)=0$
$tan~x=1$ or $tan~x=\frac{1}{2}$
$tan~x=1$
$x=\frac{\pi}{4}$
$tan~x=\frac{1}{2}$
$x=arctan~\frac{1}{2}$
The period of $tan~x$ is $\pi$. The general solutions are:
$x=\frac{\pi}{4}+n\pi$
$x=arctan~\frac{1}{2}+n\pi$
where $n$ is an integer.
