Question

If ${\tan }^{-1}x+{\tan }^{-1}y+{\tan }^{-1}z=\pi$ then prove that:
$x+y+z=xyz$

Answer 1

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

To prove: $x+y+z=xyz$

Consider ${\tan }^{-1}x+{\tan }^{-1}y+{\tan }^{-1}z=\pi$

We know that, ${\tan }^{-1}x+{\tan }^{-1}y={\tan }^{-1}\frac{x+y}{1-xy}...\left(1\right)$

$\Rightarrow {\tan }^{-1}\frac{x+y}{1-xy}+{\tan }^{-1}z=\pi$

Again use $\left(1\right)$ we get

$\Rightarrow {\tan }^{-1}\frac{\frac{x+y}{1-xy}+z}{1-\frac{x+y}{1-xy}z}=\pi$

$\Rightarrow {\tan }^{-1}\frac{\frac{x+y+z-xyz}{1-xy}}{\frac{1-xy-xz-yz}{1-xy}}=\pi$

$\Rightarrow {\tan }^{-1}\frac{x+y+z-xyz}{1-xy-yz-xz}=\pi$

$\Rightarrow \frac{x+y+z-xyz}{1-xy-yz-xz}=\tan \pi$

$\Rightarrow \frac{x+y+z-xyz}{1-xy-yz-xz}=0$

$\Rightarrow x+y+z-xyz=0$

$\Rightarrow x+y+z=xyz$

Hence proved.