Question

The number of positive integral solutions of the equation

$\tan^{- 1}x + \cos^{- 1}\frac{y}{\sqrt{1 + y^{2}}} = \sin^{- 1}\frac{3}{\sqrt{10}}$

or$\tan^{- 1}x + \cot^{- 1}y = \tan^{- 1}3$ is

• A. One
• B. Two
• C. Zero
• D. None

Answer 1

Answer: (B)

Two

Answer 2

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

$\tan^{- 1}\frac{1}{y} = \tan^{- 1}\frac{3 - x}{1 + 3x}$ ⇒ $y = \frac{1 + 3x}{3 - x}$

As $x,$ y are positive integers, $x = 1$, 2 and corresponding $y = 2,$ 7

$\therefore$ Solutions are $(x,y) = (1,2),(2,7)$