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If $A+B+C=\pi / 2$, then what is the value of $\tan A \tan B+\tan B \tan C+\tan C \tan A ?
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Given $A+B+C=\frac{\pi}{2}$ Take tan on both sides,
$\Rightarrow \tan (A+B+C)=\tan \left(\frac{\pi}{2}\right)$
$\Rightarrow \frac{\tan A+\tan B+\tan C-\tan A \tan B \tan C}{1-\tan A \tan B-\tan B \tan C-\tan C \tan A}=\frac{1}{0}$
$\left(\because \frac{1}{0}=\infty\right)$
$\Rightarrow \tan A \tan B+\tan B \tan C+\tan C \tan A=1$
$\Rightarrow \tan (A+B+C)=\tan \left(\frac{\pi}{2}\right)$
$\Rightarrow \frac{\tan A+\tan B+\tan C-\tan A \tan B \tan C}{1-\tan A \tan B-\tan B \tan C-\tan C \tan A}=\frac{1}{0}$
$\left(\because \frac{1}{0}=\infty\right)$
$\Rightarrow \tan A \tan B+\tan B \tan C+\tan C \tan A=1$
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