Question

If $x =\frac{\sqrt{3}}{3}$ and $( x +1)( y +1)=2$, then the value of $\tan ^{-1} y$ is equal to

• A. $ \frac{\pi}{12}$
• B. $\frac{\pi}{4}$
• C. $\frac{\pi}{3}$
• D. $\frac{\pi}{6}$

Answer 1

Answer: (A)

$ x=\frac{1}{\sqrt{3}}$ and $x y+x+y+1=2$
$x y+x+y=1 $
$y(1+x)=1-x$
$y=\frac{1-x}{1+x}=\frac{1-\frac{1}{\sqrt{3}}}{1+\frac{1}{\sqrt{3}}}=\frac{\sqrt{3}-1}{\sqrt{3}+1} \Rightarrow y=\tan \frac{\pi}{12} $
$\therefore \tan ^{-1} y=\frac{\pi}{12}$