Question
If $A=\left[\begin{array}{cc}\cos \alpha & \sin \alpha \\ -\sin \alpha & \cos \alpha\end{array}\right]$ show that $A^2==\left[\begin{array}{cc}\cos 2 \alpha & \sin 2 \alpha \\ -\sin 2 \alpha & \cos 2 \alpha\end{array}\right]$
$\begin{aligned} & =\left[\begin{array}{cc}\cos ^2 \alpha-\sin ^2 \alpha & \cos \alpha \sin \alpha+\cos \alpha \sin \alpha \\ -\cos \alpha \sin \alpha-\cos \alpha \sin \alpha & -\sin ^2 \alpha+\cos ^2 \alpha\end{array}\right] \\ & =\left[\begin{array}{cc}\cos ^2 \alpha-\sin ^2 \alpha & 2 \sin \alpha \cos \alpha \\ -2 \sin \alpha \cos \alpha & \cos ^2 \alpha-\sin ^2 \alpha\end{array}\right] \\ & =\left[\begin{array}{cc}\cos 2 \alpha & \sin 2 \alpha \\ -\sin 2 \alpha & \cos 2 \alpha\end{array}\right]\end{aligned}$
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