$\because {\cot }^{-1}\mathrm{\alpha }=\theta \Rightarrow \alpha =\cot \theta ...........\left(\mathrm{i}\right)$
Also, ${\mathrm{c}\mathrm{o}\mathrm{t}}^{-1}\beta =\phi \Rightarrow \beta =\cot \phi ...........\left(\mathrm{i}\mathrm{i}\right)$
Now, given $\alpha \beta =-1$

So, from $\left(\mathrm{i}\right) \& \left(\mathrm{ii}\right)$, we get,
$\cot \theta .\cot \phi =-1\Rightarrow \tan \theta \tan \phi =-1\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}.........\left(\mathrm{i}\mathrm{i}\mathrm{i}\right)$
$\mathrm{t}\mathrm{a}\mathrm{n}\left(\theta -\phi \right)=\frac{\tan \theta -\tan \phi }{1+\tan \theta \tan \phi }............\left(\mathrm{i}\mathrm{v}\right)$
$\theta -\varphi =\pm \frac{\pi }{2} (\mathrm{from} (\mathrm{iii}) \mathrm{and} (\mathrm{iv}\left)\right)$
But, $\alpha >\beta \Rightarrow \theta <\phi \Rightarrow \theta -\phi =-\frac{\pi }{2}$