If $A+B+C=\pi$, then find the value of $\begin{vmatrix} \sin \left(A+B+C\right) & \sin B & \cos C \\ - \sin B & 0 & \tan A \\ \cos \left(A+B\right) & -\tan A & 0 \end{vmatrix}$
Since $A+B+C = \pi$, $\sin \left(A+B+C\right) = \sin \pi = 0$ and $\cos \left(A+B\right) = \cos \left(\pi-C\right) = -\cos C$
$
\begin{aligned}
\text{Let } \Delta & = \begin{vmatrix}
\sin \left(A+B+C\right) & \sin B & \cos C \\
- \sin B & 0 & \tan A \\
\cos \left(A+B\right) & -\tan A & 0
\end{vmatrix} \\
& \\
& = \begin{vmatrix}
0 & \sin B & \cos C \\
-\sin B & 0 & \tan A \\
-\cos C & -\tan A & 0
\end{vmatrix} \\
& \\
& = -\sin B \times \left[0- \tan A \times \left(-\cos C\right)\right] + \cos C \sin B \tan A \\
& \\
& = -\tan A \sin B \cos C + \tan A \sin B \cos C = 0
\end{aligned}
$