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If $\tan ^{-1} x+\tan ^{-1} y+\tan ^{-1} z=\frac{\pi}{2}, \quad x, y, z>0, x y < 1, \quad$ then the value of
$x y+y z+z x=$
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$x y+y z+z x=$
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$\tan ^{-1} x+\tan ^{-1} y+\tan ^{-1} z=\frac{\pi}{2}$
$\therefore\left(\tan ^{-1} x+\tan ^{-1} y\right)=\left(\frac{\pi}{2}-\tan ^{-1} z\right)$
$\therefore \tan ^{-1}\left(\frac{x+y}{1-x y}\right)=\cot ^{-1} z=\tan ^{-1}\left(\frac{1}{z}\right)$
$\therefore \frac{x+y}{1-x y}=\frac{1}{z} \Rightarrow x z+y z=1-x y$
$\therefore x y+y z+z x=1$
$\therefore\left(\tan ^{-1} x+\tan ^{-1} y\right)=\left(\frac{\pi}{2}-\tan ^{-1} z\right)$
$\therefore \tan ^{-1}\left(\frac{x+y}{1-x y}\right)=\cot ^{-1} z=\tan ^{-1}\left(\frac{1}{z}\right)$
$\therefore \frac{x+y}{1-x y}=\frac{1}{z} \Rightarrow x z+y z=1-x y$
$\therefore x y+y z+z x=1$
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