Simplify cos theta begin{bmatrix} cos theta & | vedclass

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Question

(A) Given expression: $\cos 	heta \begin{bmatrix} \cos 	heta & \sin 	heta \ -\sin 	heta & \cos 	heta \end{bmatrix} + \sin 	heta \begin{bmatrix}

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$\begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix}$

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(A) Given expression: $\cos 	heta \begin{bmatrix} \cos 	heta & \sin 	heta \ -\sin 	heta & \cos 	heta \end{bmatrix} + \sin 	heta \begin{bmatrix} \sin 	heta & -\cos 	heta \ \cos 	heta & \sin 	heta \end{bmatrix}$ Multiply the scalar into the matrices: $= \begin{bmatrix} \cos^2 	heta & \cos 	heta \sin 	heta \ -\sin 	heta \cos 	heta & \cos^2 	heta \end{bmatrix} + \begin{bmatrix} \sin^2 	heta & -\sin 	heta \cos 	heta \ \sin 	heta \cos 	heta & \sin^2 	heta \end{bmatrix}$ Add the two matrices: $= \begin{bmatrix} \cos^2 	heta + \sin^2 	heta & \cos 	heta \sin 	heta - \sin 	heta \cos 	heta \ -\sin 	heta \cos 	heta + \sin 	heta \cos 	heta & \cos^2 	heta + \sin^2 	heta \end{bmatrix}$ Using the identity $\cos^2 	heta + \sin^2 	heta = 1$: $= \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix}$

Simplify $\cos \theta \begin{bmatrix} \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{bmatrix} + \sin \theta \begin{bmatrix} \sin \theta & -\cos \theta \\ \cos \theta & \sin \theta \end{bmatrix}$.

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