Determinant of Plane Rotation Matrix

Theorem
The matrices associated with anticlockwise and clockwise rotations of the plane about the origin through an angle of $\alpha$ both have a determinant of $1$:


 * $\begin{vmatrix}

\cos \alpha & -\sin \alpha \\ \sin \alpha & \cos \alpha \end{vmatrix} = \begin{vmatrix} \cos \alpha & \sin \alpha \\ -\sin \alpha & \cos \alpha \end{vmatrix} = 1$

Proof
The determinant of an anticlockwise plane rotation matrix is:

The determinant of a clockwise plane rotation matrix is:

Hence the result.