Question

If $A,$ $B$ and $C$ are the angles of a non-right angled triangle $ABC,$ then the value of $\left|\begin{array}{ccc}\tan A& 1& 1\\ 1& \tan B& 1\\ 1& 1& \tan C\end{array}\right|$ is equal to

• A. $1$
• B. $2$
• C. $-1$
• D. $-2$

Answer 1

The correct option is B $2$

$\left|\begin{array}{ccc}\tan A& 1& 1\\ 1& \tan B& 1\\ 1& 1& \tan C\end{array}\right|$

$=\tan A(\tan B\tan C-1)-(\tan C-1)+(1-\tan B)$
$=\tan A\tan B\tan C-\tan A-\tan B-\tan C+2$
$=2$ as $\tan A+\tan B+\tan C=\tan A\tan B\tan C$