Question

If $A+B+C=\pi$, then $\left|\begin{array}{ccc}sin(A+B+C)& sinB& cosC\\ -sinB& 0& tanA\\ cos(A+B)& -tanA& 0\end{array}\right|=$ .............

Answer 1

$A+B+C=\pi$

$\Rightarrow \sin 180=0\cos (A+B)=\cos (180-C)=-\cos C$

$\Rightarrow \left|\begin{array}{ccc}0& \sin B& \cos C\\ \sin B& 0& \tan A\\ -\cos C& -\tan A& 0\end{array}\right|$

The obtained matrix is a skew symmetric

$det=0$