If ${\tan ^{ - 1}}x + {\tan ^{ - 1}}y = \frac{{4\pi }}{5}$, then ${\cot ^{ - 1}}x + {\cot ^{ - 1}}y$ equals to

We have, ${\tan ^{ - 1}}x + {\tan ^{ - 1}}y = \frac{{4\pi }}{5}$

$\Rightarrow$ $\frac{\pi }{2} - {\cot ^{ - 1}}x + \frac{\pi }{2} - {\cot ^{ - 1}}y = \frac{{4\pi }}{5}$

$\Rightarrow$ $- \left( {{{\cot }^{ - 1}}x + {{\cot }^{ - 1}}y} \right) = \frac{{4\pi }}{5} - \pi$

$\Rightarrow$ ${\cot ^{ - 1}}x + {\cot ^{ - 1}}y = - \left( { - \frac{\pi }{5}} \right)$

$\Rightarrow$ ${\cot ^{ - 1}}x + {\cot ^{ - 1}}y = \frac{\pi }{5}$