Question

If $△ABC$ is not a right triangle, then value of
$\Delta =\left|\begin{array}{ccc}\tan A& 1& 1\\ 1& \tan B& 1\\ 1& 1& \tan C\end{array}\right|$
is

• A. $-1$
• B. $2$
• C. $3$
• D. $0$

Answer 1

The correct option is B $2$

In $\Delta ABC$
We can use the below stated standard for the triangle $ABC$, the result can be proved easily by using sum and difference angles formula for $\tan$
$A+B+C=\pi \Rightarrow \tan A+\tan B+\tan C=\tan A\tan B\tan C$
Thus
$\Delta =\left|\begin{array}{ccc}\tan A& 1& 1\\ 1& \tan B& 1\\ 1& 1& \tan C\end{array}\right|=\tan A(\tan B\tan C-1)-1(\tan C-1)+(1-\tan B)=\tan A\tan B\tan C-\tan A-\tan C+1+1-\tan B=2$