Prove that `tan^(-1) x + tan^(-1).(1)/(x) = {(pi//2,"if" x gt 0),(-pi//2," if " x lt 0):}`

Explanation

To prove that

\[
\tan^{-1} x + \tan^{-1} \left(\frac{1}{x}\right) =
\begin{cases}
\frac{\pi}{2} & \text{if } x > 0 \\
-\frac{\pi}{2} & \text{if } x < 0
\end{cases}
...