If $\alpha \gt \beta \gt \gamma \gt 0$, then the expression $\cot ^{-1}\left\{\beta+\frac{\left(1+\beta^2\right)}{(\alpha-\beta)}\right\}+\cot ^{-1}\left\{\gamma+\frac{\left(1+\gamma^2\right)}{(\beta-\gamma)}\right\}+$ $\cot…

Question

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

Aicharya · Accepted Answer

$\begin{aligned} & \Rightarrow \cot ^{-1}\left(\frac{\alpha \beta+1}{\alpha-\beta}ight)+\cot ^{-1}\left(\frac{\beta \gamma+1}{\beta-\gamma}ight)+\cot ^{-1}\left(\frac{\alpha \gamma+1}{\gamma-\alpha}ight) \ & \Rightarrow 	an ^{-1}\left(\frac{\alpha-\beta}{1+\alpha \beta}ight)+	an ^{-1}\left(\frac{\beta-\gamma}{1+\beta \gamma}ight)+\pi+	an ^{-1}\left(\frac{\gamma-\alpha}{1+\gamma \alpha}ight) \ & \Rightarrow\left(	an ^{-1} \alpha-	an ^{-1} \betaight)+\left(	an ^{-1} \beta-	an ^{-1} \gammaight)+\left(\pi+	an ^{-1} \gamma-	an ^{-1} \alphaight) \ & \Rightarrow \pi\end{aligned}$ .

If $\alpha \gt \beta \gt \gamma \gt 0$, then the expression $\cot…

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