If $x0, y0, z0, x y+y z+z x

Question

If $x>0, y>0, z>0, x y+y z+z x < 1$ and if $	an ^{-1} x+	an ^{-1} y+	an ^{-1} z=\pi$, then $x+y+z$ equals to

Aicharya · Accepted Answer

$\begin{aligned} & 	ext { Given, } 	an ^{-1} x+	an ^{-1} y+	an ^{-1} z=\pi \ & \Rightarrow \quad 	an ^{-1}\left(\frac{x+y+z-x y z}{1-x y-y z-z x}ight)=\pi \ & \Rightarrow \quad \frac{x+y+z-x y z}{1-x y-y z-z x}=	an \pi=0 \ & \Rightarrow \quad x+y+z=x y z\end{aligned}$

If $x>0, y>0, z>0, x y+y z+z x < 1$ and if $\tan ^{-1} x+\tan ^{-1} y+\tan ^{-1} z=\pi$, then $x+y+z$ equals…

Solution