Question

Find all positive integral solution of the equation ${\tan }^{-1}x+{\cot }^{-1}y={\tan }^{-1}3$

Answer 1

We have to find values of $x$ and $y$ which are positive as well as integer

Given:${\tan }^{-1}x+{\cot }^{-1}y={\tan }^{-1}3$

$\Rightarrow {\tan }^{-1}x+{\tan }^{-1}\frac{1}{y}={\tan }^{-1}3$ where $y>0$

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

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

$\Rightarrow \frac{x+\frac{1}{y}}{1-\frac{x}{y}}=3$

$\Rightarrow 3x+\frac{3}{y}=1-\frac{x}{y}$

$\Rightarrow 3x+1=y(3-x)$

$\Rightarrow y=\frac{3x+1}{3-x}$

when $x$ is positive, numerator is positive for $y$ to be positive denominator must be positive

i.e., $3–x>0$ or $x<3\Rightarrow x=1,2$

for $x=1\Rightarrow y=2$

for $x=2\Rightarrow y=7$