If $\mathrm{\alpha }>\mathrm{\beta }>\mathrm{\gamma }>0$, then the expression 
${\cot }^{-1}\left\{\mathrm{\beta }+\frac{\left(1+{\mathrm{\beta }}^2\right)}{(\mathrm{\alpha }-\mathrm{\beta })}\right\}+{\cot }^{-1}\left\{\mathrm{\gamma }+\frac{\left(1+{\mathrm{\gamma }}^2\right)}{(\mathrm{\beta }-\mathrm{\gamma })}\right\}+{\cot }^{-1}\left\{\mathrm{\alpha }+\frac{\left(1+{\mathrm{\alpha }}^2\right)}{(\mathrm{\gamma }-\mathrm{\alpha })}\right\}$ is equal to : 

$\begin{array}{l}\Rightarrow {\cot }^{-1}\left(\frac{\mathrm{\alpha \beta }+1}{\mathrm{\alpha }-\mathrm{\beta }}\right)+{\cot }^{-1}\left(\frac{\mathrm{\beta \gamma }+1}{\mathrm{\beta }-\mathrm{\gamma }}\right)+{\cot }^{-1}\left(\frac{\mathrm{\alpha \gamma }+1}{\mathrm{\gamma }-\mathrm{\alpha }}\right)\\ \Rightarrow {\tan }^{-1}\left(\frac{\mathrm{\alpha }-\mathrm{\beta }}{1+\mathrm{\alpha \beta }}\right)+{\tan }^{-1}\left(\frac{\mathrm{\beta }-\mathrm{\gamma }}{1+\mathrm{\beta \gamma }}\right)+\mathrm{\pi }+{\tan }^{-1}\left(\frac{\mathrm{\gamma }-\mathrm{\alpha }}{1+\mathrm{\gamma \alpha }}\right)\\ \Rightarrow \left({\tan }^{-1}\mathrm{\alpha }-{\tan }^{-1}\mathrm{\beta }\right)+\left({\tan }^{-1}\mathrm{\beta }-{\tan }^{-1}\mathrm{\gamma }\right)+\left(\mathrm{\pi }+{\tan }^{-1}\mathrm{\gamma }-{\tan }^{-1}\mathrm{\alpha }\right)\\ \Rightarrow \mathrm{\pi }\end{array}$