How to solve this trig equation , $\tan^{-1}(x) = 1 / \tan (x)$?

Question

Given the equation $$\tan^{-1}(x) = \frac{1}{\tan(x)},\quad x\in[0,2\pi],$$ find the value/values of $x$.

I tried to take $\tan (x)$ for the both sides but the equation had more complicated !

Answer 1

I believe you will not be able to find exact solutions, only numerical. However, as you note, it does have solutions, it has $2n$ solutions in every interval of the form $[-n \pi, n \pi]$.

Answer 2

There are two solutions in the given interval and you can find them only numerically.