If ^ -1 x+ ^ -1 y+ ^ -1 z= 2 , then - Vidyadip

MCQ

If $\tan ^{-1} x+\tan ^{-1} y+\tan ^{-1} z=\frac{\pi}{2}$, then

A
$x+y+z-x y z=0$
B
$x+y+z+x y z=0$
C
$x y+y z+z x+1=0$
✓
$x y+y z+z x-1=0$

✓

Answer

Correct option: D.

$x y+y z+z x-1=0$

(D) $\tan ^{-1} x+\tan ^{-1} y+\tan ^{-1} z=\frac{\pi}{2}$
$\Rightarrow \tan ^{-1}[\frac{x+y+z-x y z}{1-x y-y z-x z}]=\frac{\pi}{2}$ ...[Using Shortcut 4]
$\Rightarrow[\frac{x+y+z-x y z}{1-x y-y z-z x}]=\tan \frac{\pi}{2}$
$\Rightarrow x y+y z+z x-1=0$
Alternate Method:
Let $x=y= z =\frac{1}{\sqrt{3}}$
Then, $\tan ^{-1} \frac{1}{\sqrt{3}}+\tan ^{-1} \frac{1}{\sqrt{3}}+\tan ^{-1} \frac{1}{\sqrt{3}}=\frac{\pi}{2}$
Option (D) holds for these values of $x, y, z$

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