Question
Simplify $\cos \theta\left[\begin{array}{cc}\cos \theta & \sin \theta \\ -\sin \theta & \cos \theta\end{array}\right]+\sin \theta\left[\begin{array}{cc}\sin \theta & -\cos \theta \\ \cos \theta & \sin \theta\end{array}\right]$

Answer

$\cos \theta\left[\begin{array}{cc}\cos \theta & \sin \theta \\ -\sin \theta & \cos \theta\end{array}\right]+\sin \theta\left[\begin{array}{cc}\sin \theta & -\cos \theta \\ \cos \theta & \sin \theta\end{array}\right]$

$=\left[\begin{array}{cc}\cos ^2 \theta & \cos \theta \sin \theta \\ -\cos \theta \sin \theta & \cos ^2 \theta\end{array}\right]+\left[\begin{array}{cc}\sin ^2 \theta & -\cos \theta \sin \theta \\ \cos \theta \sin \theta & \sin ^2 \theta\end{array}\right]$

$=\left[\begin{array}{cc}\cos ^2 \theta+\sin ^2 \theta & \cos \theta \sin \theta-\cos \theta \sin \theta \\ -\cos \theta \sin \theta+\cos \theta \sin \theta & \cos ^2 \theta+\sin ^2 \theta\end{array}\right]$

$=\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]$

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