Question

The value of $\tan ^{-1}\left(\sqrt{\frac{x(x+y+z)}{y z}}\right)+\tan ^{-1}\left(\sqrt{\frac{y(x+y+z)}{z x}}\right)+\tan ^{-1}\left(\sqrt{\frac{z(x+y+z)}{x y}}\right)$ is (where $x , y , z$ are distinct positive real numbers)

• A. $ \pi$
• B. $\frac{\pi}{4}$
• C. $\frac{\pi}{2}$
• D. $\frac{3 \pi}{4}$

Answer 1

Answer: (A)

Let $a=\sqrt{\frac{x+y+z}{x y z}} $
$\tan ^{-1}(a x)+\tan ^{-1}(a y)+\tan ^{-1}(a z)=\tan ^{-1}\left(\frac{a x+a y+a z-a^3 x y z}{1-()}\right)$
$=\tan ^{-1}\left(\frac{a\left(x+y+z-a^2 x y z\right)}{d r}\right)=\tan ^{-1}\left(\frac{0}{d r}\right)=\pi$ .