Question

If $\alpha ϵ(-\frac{\pi }{2},0)$, then find the value of ${\tan }^{-1}\left(\cot \alpha \right)-{\cot }^{-1}\left(\tan \alpha \right)$

Answer 1

In the interval of $(\frac{-\pi }{2},0)$ both $tan\alpha$ and $cot\alpha$ are negative.
Now
$ta{n}^{-1}\left(cot\alpha \right)-co{t}^{-1}\left(tan\alpha \right)$
$=\frac{\pi }{2}-co{t}^{-1}\left(cot\alpha \right)-\frac{\pi }{2}+ta{n}^{-1}\left(tan\alpha \right)$
$=ta{n}^{-1}\left(tan\alpha \right)-co{t}^{-1}\left(cot\alpha \right)$
$=\alpha -(\pi +\alpha )$
$=-\pi$.