Question

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

• A. $0$
• B. $xyz$
• C. $3xyz$
• D. $\sqrt{xyz}$

Answer 1

Answer: (B)

$xyz$

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

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

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

$\Rightarrow x+y=-z(1-xy)\Rightarrow x+y+z=xyz$