Question:
If A + B = C, then write the value of tan A tan B tan C.
Solution:
$\tan A \tan B \tan C=\tan A \tan B \tan (A+B)$ $[$ Using $A+B=C]$
$=\tan A \tan B \times \frac{\tan A+\tan B}{1-\tan A \tan B}$
$=\frac{\tan ^{2} A \tan B+\tan A \tan ^{2} B}{1-\tan A \tan B}$
$=\frac{\tan ^{2} A \tan B+\tan A \tan ^{2} B+\tan A+\tan B-\tan A-\tan B}{1-\tan A \tan B}$
$=\frac{-\tan A(1-\tan A \tan B)-\tan B(1-\tan A \tan B)+\tan A+\tan B}{1-\tan A \tan B}$
$=\frac{-(1-\tan A \tan B)(\tan A+\tan B)+\tan A+\tan B}{1-\tan A \tan B}$
$=\frac{\tan A+\tan B}{1-\tan A \tan B}-\tan A-\tan B$
$=\tan (A+B)-\tan A-\tan B$
$=\tan C-\tan A-\tan B$
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