Question

If xy + yz + zx = 1, then :

• A. $\tan^{-1} x + \tan^{-1} y + \tan^{-1} z = 0 $
• B. $\tan^{-1} x + \tan^{-1} y + \tan^{-1} z = \pi $
• C. $\tan^{-1} x + \tan^{-1} y + \tan^{-1} z = \frac{\pi}{4} $
• D. $\tan^{-1} x + \tan^{-1} y + \tan^{-1} z = \frac{\pi}{2} $

Answer 1

Answer: (D)

$xy + yz + zx = 1$ ......(1)
Now, we know $ \tan ^{-1} x + \tan ^{-1} y + \tan ^{-1} z $
$= \tan^{-1} \left[ \frac{x+y+z-xyz}{1-\left(xy+yz+zx\right)}\right]$
using equation (1) we have
$ \tan ^{-1} x + \tan ^{-1} y + \tan ^{-1} z $
$ = \tan^{-1} \left(\frac{1}{0}\right) = \tan\infty $
$ \Rightarrow \tan^{-1} x + \tan^{-1} y + \tan^{-1} z = \frac{\pi}{2} $