Show that for every matrix $A$ of $\mathrm{O}(2)$ such that $\det(A) = -1$, there exists a real $t$ such that $$A = \left(\begin{array}{cr} \cos(2t) & \sin(2t) \\ \sin(2t) & -\cos(2t) \end{array}\right)$$
Show that for every matrix $A$ of $\mathrm{O}(2)$ such that $\det(A) = -1$, there exists a real $t$ such that
$$A = \left(\begin{array}{cr} \cos(2t) & \sin(2t) \\ \sin(2t) & -\cos(2t) \end{array}\right)$$