if

We know

$\tan ^{-1} a+\cot ^{-1} a=\frac{\pi}{2}$, for all $a \in \mathrm{R}$           .....(1)

Now, 

$\tan ^{-1} x+\tan ^{-1} y=\frac{4 \pi}{5}$                      (Given)

$\Rightarrow \frac{\pi}{2}-\cot ^{-1} x+\frac{\pi}{2}-\cot ^{-1} y=\frac{4 \pi}{5}$       [Using (1)]

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

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

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