Evaluate

$\Delta=\left|\begin{array}{ccc}\cos \alpha \cos \beta & \cos \alpha \sin \beta & -\sin \alpha \\ -\sin \beta & \cos \beta & 0 \\ \sin \alpha \cos \beta & \sin \alpha \sin \beta & \cos \alpha\end{array}\right|$

Expanding along $\mathrm{C}_{3}$, we have:

$\begin{aligned} \Delta &=-\sin \alpha\left(-\sin \alpha \sin ^{2} \beta-\cos ^{2} \beta \sin \alpha\right)+\cos \alpha\left(\cos \alpha \cos ^{2} \beta+\cos \alpha \sin ^{2} \beta\right) \\ &=\sin ^{2} \alpha\left(\sin ^{2} \beta+\cos ^{2} \beta\right)+\cos ^{2} \alpha\left(\cos ^{2} \beta+\sin ^{2} \beta\right) \\ &=\sin ^{2} \alpha(1)+\cos ^{2} \alpha(1) \\ &=1 \end{aligned}$