If αgtβgtγgt0, then the expression cot ^-1 β+ (1+β^2 ) (α-β) +cot ^-1 γ+ (1+γ^2 ) (β-γ) +cot ^-1 α+ (1+α^2 ) (γ-α) is equal to :

$$\begin{aligned} & \Rightarrow \cot ^{-1}\left(\frac{\alpha \beta+1}{\alpha-\beta}\right)+\cot ^{-1}\left(\frac{\beta \gamma+1}{\beta-\gamma}\right)+\cot ^{-1}\left(\frac{\alpha \gamma+1}{\gamma-\alpha}\right) \\ & \Rightarrow \tan ^{-1}\left(\frac{\alpha-\beta}{1+\alpha \beta}\right)+\tan ^{-1}\left(\frac{\beta-\gamma}{1+\beta \gamma}\right)+\pi+\tan ^{-1}\left(\frac{\gamma-\alpha}{1+\gamma \alpha}\right) \\ & \Rightarrow\left(\tan ^{-1} \alpha-\tan ^{-1} \beta\right)+\left(\tan ^{-1} \beta-\tan ^{-1} \gamma\right)+\left(\pi+\tan ^{-1} \gamma-\tan ^{-1} \alpha\right) \\ & \Rightarrow \pi\end{aligned}$$​⇒cot−1(α−βαβ+1​)+cot−1(β−γβγ+1​)+cot−1(γ−ααγ+1​)⇒tan−1(1+αβα−β​)+tan−1(1+βγβ−γ​)+π+tan−1(1+γαγ−α​)⇒(tan−1α−tan−1β)+(tan−1β−tan−1γ)+(π+tan−1γ−tan−1α)⇒π​