Question

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

Answer 1

Given that $\tan ^{-1} x+\tan ^{-1} y+\tan ^{-1} z=\frac{\pi}{2}$

$\text { then } \tan ^{-1}\left[\frac{x+y+z-x y z}{1-x y-y z-z x}\right]=\frac{\pi}{2}$

$\Rightarrow \quad \frac{x+y+z-x y z}{1-x y-y z-z x}=\tan \frac{\pi}{2}=\infty$

$\Rightarrow \quad 1-x y-y z-z x=0$

$\Rightarrow \quad x y+y z+z x=1$