Question

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

• A. $\frac{3\pi }{5}$
• B. $\pi$
• C. $\frac{2\pi }{5}$
• D. $\frac{\pi }{5}$

Answer 1

The correct option is D $\frac{\pi }{5}$

Given:

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

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

$\because {\tan }^{-1}x+{\cot }^{-1}x=\frac{\pi }{2}$

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

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

Therefore, option $\left(a\right)$ is correct