$$\begin{gathered} \mathrm{lf} \mathrm{\omega } =\sqrt{{\mathrm{z}}^2-1},\mathrm{where} \mathrm{\omega } \mathrm{is} \mathrm{a} \mathrm{cube} \mathrm{root} \mathrm{of} \mathrm{unity} \\ \mathrm{and} \mathrm{z}=\mathrm{x}+\mathrm{iy} \mathrm{then} \\ \frac{1}{2}\left[{\mathrm{Tan}}^{-1}\left(\frac{\mathrm{y}}{\mathrm{x}+1}\right)+{\mathrm{Tan}}^{-1}\left(\frac{\mathrm{y}}{\mathrm{x}-1}\right)\right]= \end{gathered}$$

$\begin{array}{rcl}\mathrm{w}& =& \sqrt{{\mathrm{z}}^2-1}, \mathrm{z}=\mathrm{x}+\mathrm{iy} \Rightarrow {\mathrm{w}}^2={(\mathrm{x}+\mathrm{iy})}^2-1 \\ & & \end{array}$

$\begin{array}{rcl} & \Rightarrow & \frac{-1-\mathrm{i}\sqrt{3}}{2} =\left({\mathrm{x}}^2-{\mathrm{y}}^2-1\right)+\mathrm{i}2\mathrm{xy}\end{array}$

$\begin{array}{rcl} & \Rightarrow & {\mathrm{x}}^2-{\mathrm{y}}^2-1=\frac{-1}{2}\mathrm{and}\end{array} \begin{array}{rcl} & & 2\end{array}\begin{array}{rcl} & & \mathrm{xy}\end{array}\begin{array}{rcl} & =& \end{array}\begin{array}{rcl} & & \frac{-\sqrt{3}}{2}\end{array}$

$\frac{1}{2}{\tan }^{-1}\left(\frac{\mathrm{y}}{\mathrm{x}+1}\right)+{\tan }^{-1}\left(\frac{\mathrm{y}}{\mathrm{x}-1}\right)$

$\frac{1}{2}{\tan }^{-1}\left(\frac{{\displaystyle \frac{\mathrm{y}}{\mathrm{x}+1}+\frac{\mathrm{y}}{\mathrm{x}-1}}}{1-{\displaystyle \frac{\mathrm{y}}{\mathrm{x}+1}}·{\displaystyle \frac{\mathrm{y}}{\mathrm{x}-1}}}\right)$

$=\frac{1}{2}{\tan }^{-1}\frac{2\mathrm{xy}}{{\mathrm{x}}^2-{\mathrm{y}}^2-1}$

$=\frac{1}{2} {\tan }^{-1} \frac{{\displaystyle \raisebox{1ex}{$-\sqrt{3}$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}{{\displaystyle \raisebox{1ex}{$-1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}$

$=\frac{1}{2}{\tan }^{-1}\sqrt{3}=\frac{1}{2}\left(\frac{\mathrm{\pi }}{3}\right)=\frac{\mathrm{\pi }}{6}$