Question

The number of positive integral solutions of ${\tan }^{-1}x+{\cos }^{-1}\frac{y}{\sqrt{1+y^2}}={\sin }^{-1}\frac{3}{\sqrt{10}}$ is

• A. $1$
• B. $2$
• C. $3$
• D. $4$

Answer 1

The correct option is B $2$

Here, ${\tan }^{-1}x+{\cos }^{-1}\frac{y}{\sqrt{1-y^2}}={\sin }^{-1}\frac{3}{\sqrt{10}}$
or ${\tan }^{-1}x+{\tan }^{-1}(\frac{1}{y})={\tan }^{-1}\left(3\right)$
or ${\tan }^{-1}(\frac{1}{y})={\tan }^{-1}3-{\tan }^{-1}\left(x\right)$

or ${\tan }^{-1}(\frac{1}{y})={\tan }^{-1}(\frac{3-x}{1+3x})$

or $y=\frac{1+3x}{3-x}$
As $x,y$ are positive integers, $x=1,2$ and corresponding $y=2,7.$
Therefore, the solutions are $(x,y)=(1,2)\text{and}(2,7)$ i.e., there are two solutions.