Question:
Evaluate $\Delta=\left|\begin{array}{ccc}0 & \sin \alpha & -\cos \alpha \\ -\sin \alpha & 0 & \sin \beta \\ \cos \alpha & -\sin \beta & 0\end{array}\right|$
Solution:
Let $\Delta=\left|\begin{array}{ccc}0 & \sin \alpha & -\cos \alpha \\ -\sin \alpha & 0 & \sin \beta \\ \cos \alpha & -\sin \beta & 0\end{array}\right|$
$\Delta=(-1)^{1+1} 0\left(0+\sin ^{2} \beta\right)+(-1)^{1+2} \sin \alpha(0-\sin \beta \cos \alpha)+(-1)^{1+3}(-\cos \alpha)(\sin \alpha \sin \beta-0) \quad\left[\right.$ Expanding along $\left.R_{1}\right]$
$=0\left(0+\sin ^{2} \beta\right)-\sin \alpha(0-\sin \beta \cos \alpha)-\cos \alpha(\sin \alpha \sin \beta-0)$
$=\sin \alpha \sin \beta \cos \alpha-\sin \alpha \sin \beta \cos \alpha$
$=0$