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