If A + B = C, Find tan A tan B tan C

Solution

Using, \[ \tan(A+B) = \frac{\tan A+\tan B}{1-\tan A\tan B} \]

Since \[ A+B=C \]

\[ \tan C = \frac{\tan A+\tan B}{1-\tan A\tan B} \]

\[ \tan C(1-\tan A\tan B) = \tan A+\tan B \]

\[ \tan C = \tan A+\tan B+\tan A\tan B\tan C \]

Therefore, \[ \tan A\tan B\tan C = \tan C-\tan A-\tan B \]

\[ \boxed{ \tan A\tan B\tan C = \tan C-\tan A-\tan B } \]