Question

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

Answer 1

$A+B=\pi -C$
$\tan (A+B)=-\tan C$
$\frac{\tan A+\tan B}{1-\tan A\tan B}=-\tan C$
$\tan A+\tan B+\tan C=\tan A\tan B\tan C$ ....(1)
Consider, $\left|\begin{array}{ccc}\tan A& 1& 1\\ 1& \tan B& 1\\ 1& 1& \tan C\end{array}\right|$
$\Delta =\tan A\tan B\tan C-\tan A-\tan C+1+1-\tan B$
$\Delta =2$ (by (1))