What is the value of $\tan^{-1}(-5) + \cot^{-1}(-5)$?

Explanation

1. 01

   Let $\theta = \tan^{-1}(-5)$, in $\left(-\frac{\pi}{2}, 0\right)$. Then $\frac{\pi}{2} - \theta$ lies in $(0, \pi)$ and $\cot\left(\frac{\pi}{2} - \theta\right) = \tan \theta = -5$.

2. 02

   So $\cot^{-1}(-5) = \frac{\pi}{2} - \theta$, and the sum is $\frac{\pi}{2}$, as it is for every real input.