Evaluate
\[\begin{vmatrix} \cos \alpha \cos \beta & \cos \alpha \sin \beta & -\sin \alpha \\ -\sin \beta & \cos \beta & 0 \\ \sin \alpha \cos \beta & \sin \alpha \sin \beta & \cos \alpha \end{vmatrix}.\]
I recognize this matrix as a rotation matrix that rotates a vector by an angle $\alpha$ around an axis given by the unit vector $(\cos \beta, \sin \beta, 0)$. I also know that the determinant of a rotation matrix is always 1, since rotation preserves the length and orientation of vectors. Therefore, without doing any calculations, I can conclude that the determinant of this matrix is 1.

