Chapter 3, Problem 54E

For any positive integer n and any angle $\theta$ , show that in the group $SL(2,R)$ , $\left[\begin{array}{l}\cos \theta -\sin \theta \\ \sin \theta \cos \theta \end{array}\right]=\left[\begin{array}{l}\cos n\theta -\sin n\theta \\ \sin n\theta \cos n\theta \end{array}\right]$ .Use this formula to find the order of $\left[\begin{array}{l}\cos 60°-\sin 60°\\ \sin 60°\cos 60°\end{array}\right]=\left[\begin{array}{l}\cos \sqrt{2°}-\sin \sqrt{2°}\\ \sin \sqrt{2°}\cos \sqrt{2°}\end{array}\right]$ .(Geometrically, $\left[\begin{array}{l}\cos \theta -\sin \theta \\ \sin \theta \cos \theta \end{array}\right]$ represents a rotation of the plane $\theta$ degrees.)