x +y + z = xyz, then the value of tan ^-1x + tan^-1y + tan^-1z = ?

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$$\begin{gathered} \mathrm{We} \mathrm{have}: \\ \mathrm{x}+\mathrm{y}+\mathrm{z}=\mathrm{xyz} \\ \Rightarrow \mathrm{x}+\mathrm{y}=\mathrm{xyz}-\mathrm{z} \\ \Rightarrow \mathrm{x}+\mathrm{y}=-\mathrm{z}\left(1-\mathrm{xy}\right) \\ \Rightarrow \frac{\mathrm{x}+\mathrm{y}}{\left(1-\mathrm{xy}\right)}=-\mathrm{z} \\ \mathrm{Applying} {\tan }^{-1} \mathrm{on} \mathrm{both} \mathrm{sides} \mathrm{we} \mathrm{get}, \\ {\tan }^{-1}\left(\frac{\mathrm{x}+\mathrm{y}}{1-\mathrm{xy}}\right)={\tan }^{-1} \left(-\mathrm{z}\right) \\ \Rightarrow {\tan }^{-1}\left(\frac{\mathrm{x}+\mathrm{y}}{1-\mathrm{xy}}\right)=-{\tan }^{-1} \mathrm{z} \\ \mathbf{Case}\left(\mathbf{i}\right)\mathbf{ }\mathbf{x}\mathbf{,}\mathbf{y}\mathbf{>}\mathbf{0}\mathbf{ }\mathbf{and}\mathbf{ }\mathbf{xy}\mathbf{<}\mathbf{1}\mathbf{,}\mathbf{ }\mathbf{then} \\ {\tan }^{-1}\left(\frac{\mathrm{x}+\mathrm{y}}{1-\mathrm{xy}}\right)=-{\tan }^{-1} \left(\mathrm{z}\right) \\ \mathrm{will} \mathrm{be} \\ {\tan }^{-1}\left(\mathrm{x}\right)+{\tan }^{-1}\left(\mathrm{y}\right)=-{\tan }^{-1}\left(\mathrm{z}\right) \\ \mathbf{\Rightarrow }{\mathbf{tan}}^{\mathbf{-}\mathbf{1}}\left(\mathbf{x}\right)\mathbf{+}{\mathbf{tan}}^{\mathbf{-}\mathbf{1}}\left(\mathbf{y}\right)\mathbf{+}{\mathbf{tan}}^{\mathbf{-}\mathbf{1}}\left(\mathbf{z}\right)\mathbf{=}\mathbf{0} \\ \mathbf{Case}\left(\mathbf{ii}\right)\mathbf{ }\mathbf{x}\mathbf{,}\mathbf{y}\mathbf{>}\mathbf{0}\mathbf{ }\mathbf{and}\mathbf{ }\mathbf{xy}\mathbf{>}\mathbf{1}\mathbf{,}\mathbf{ }\mathbf{then} \\ {\tan }^{-1}\left(\frac{\mathrm{x}+\mathrm{y}}{1-\mathrm{xy}}\right)=-{\tan }^{-1} \left(\mathrm{z}\right) \\ \mathrm{will} \mathrm{be} \\ \mathrm{\pi }+{\tan }^{-1}\left(\mathrm{x}\right)+{\tan }^{-1}\left(\mathrm{y}\right)=-{\tan }^{-1}\left(\mathrm{z}\right) \\ \mathbf{\Rightarrow }{\mathbf{tan}}^{\mathbf{-}\mathbf{1}}\left(\mathbf{x}\right)\mathbf{+}{\mathbf{tan}}^{\mathbf{-}\mathbf{1}}\left(\mathbf{y}\right)\mathbf{+}{\mathbf{tan}}^{\mathbf{-}\mathbf{1}}\left(\mathbf{z}\right)\mathbf{=}\mathbf{-}\mathbf{\pi } \\ \mathbf{Case}\left(\mathbf{iii}\right)\mathbf{ }\mathbf{x}\mathbf{,}\mathbf{y}\mathbf{<}\mathbf{0}\mathbf{ }\mathbf{and}\mathbf{ }\mathbf{xy}\mathbf{>}\mathbf{1}\mathbf{,}\mathbf{ }\mathbf{then} \\ {\tan }^{-1}\left(\frac{\mathrm{x}+\mathrm{y}}{1-\mathrm{xy}}\right)=-{\tan }^{-1} \left(\mathrm{z}\right) \\ \mathrm{will} \mathrm{be} \\ -\mathrm{\pi }+{\tan }^{-1}\left(\mathrm{x}\right)+{\tan }^{-1}\left(\mathrm{y}\right)=-{\tan }^{-1}\left(\mathrm{z}\right) \\ \mathbf{\Rightarrow }{\mathbf{tan}}^{\mathbf{-}\mathbf{1}}\left(\mathbf{x}\right)\mathbf{+}{\mathbf{tan}}^{\mathbf{-}\mathbf{1}}\left(\mathbf{y}\right)\mathbf{+}{\mathbf{tan}}^{\mathbf{-}\mathbf{1}}\left(\mathbf{z}\right)\mathbf{=}\mathbf{\pi } \end{gathered}$$

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