Question

If $\tan A+\tan B+\tan C=\tan A.\tan B.\tan C$, then

• A. A,B,C must be angles of a triangle
• B. the sum of any two of A,B,C is equal to the third
• C. A+B+C must be n integral multiple of $\pi$
• D. None of these

Answer 1

The correct option is C A+B+C must be n integral multiple of $\pi$

If $A+B+C=\pi$
$\Rightarrow \tan (A+B+C)=\tan \pi$
$\Rightarrow \frac{\tan A+\tan B+\tan C-\tan A\tan B\tan C}{1-\tan A\tan B-\tan B\tan C-\tan A\tan C}=0$
$\Rightarrow \tan A+\tan B+\tan C-\tan A\tan B\tan C=0$