Question

$A,B,C$ are three angles such that $\tan A+\tan B+\tan C=\tan A\tan B\tan C.$ Which of the following statements is always correct ?

• A. $ABC$ is a triangle, i.e. $A+B+C=\pi$
• B. $A=B=C.i.e.,$ $ABC$ is an equilateral triangle
• C. $A+B=C,$ i.e., $ABC$ is a right- angled triangle
• D. $A+B=\pi$

Answer 1

The correct option is A $ABC$ is a triangle, i.e. $A+B+C=\pi$

(A) $tan\left[\right(A+B)+C]$
$=\frac{tan(A+B)+tanC}{1-tan(A+B)tanC}=\frac{\frac{tanA+tanB}{1-tanAtanB}+tanC}{1-\frac{tanA+tanB}{1-tanAtanB}.tanC}$
$=\frac{tanA+tanB+tanC-tanAtanBtanC}{Denominator}$
$=0$
$[since,tanA+tanB+tanC=tanAtanBtanC]$
$\therefore A+B+C=\pi$ i.e., A, B, C is a triangle