Answer
See explanations.
Work Step by Step
(a) Step 1. List the matrices: $Z=\begin{bmatrix} x\\y \end{bmatrix}, Z'=\begin{bmatrix} X\\Y \end{bmatrix}, R=\begin{bmatrix} cos\phi&-sin\phi\\sin\phi&cos\phi \end{bmatrix}$
Step 2. Evaluate the products of matrices:
$RZ'=\begin{bmatrix} cos\phi&-sin\phi\\sin\phi&cos\phi \end{bmatrix}\begin{bmatrix} X\\Y \end{bmatrix}
=\begin{bmatrix} Xcos\phi-Ysin\phi\\Xsin\phi+Ycos\phi \end{bmatrix}$
Step 3. Recall the formula: $x=Xcos\phi-Ysin\phi$, $y=Xsin\phi+Ycos\phi$ for the Rotation of Axes. we have
$Z=\begin{bmatrix} x\\y \end{bmatrix}=\begin{bmatrix} Xcos\phi-Ysin\phi\\Xsin\phi+Ycos\phi \end{bmatrix}=RZ'$
Step 4. Use the inverse formula to get $R^{-1}=\frac{1}{cos^2\phi+sin^2\phi}\begin{bmatrix} cos\phi&sin\phi\\-sin\phi&cos\phi \end{bmatrix}=\begin{bmatrix} cos\phi&sin\phi\\-sin\phi&cos\phi \end{bmatrix}$
Step 5. $R^{-1}Z=\begin{bmatrix} cos\phi&sin\phi\\-sin\phi&cos\phi \end{bmatrix}\begin{bmatrix} x\\y \end{bmatrix}=\begin{bmatrix} x\cdot cos\phi+y\cdot sin\phi\\-x\cdot sin\phi+y\cdot cos\phi \end{bmatrix}$
Step 6. Compare with the formula for the Rotation of Axes, we have
$Z'=\begin{bmatrix} X\\Y \end{bmatrix}=\begin{bmatrix} x\cdot cos\phi+y\cdot sin\phi\\-x\cdot sin\phi+y\cdot cos\phi \end{bmatrix}=R^{-1}Z$
(b) Step 1. Let $R_1=\begin{bmatrix} cos\phi_1&-sin\phi_1\\sin\phi_1&cos\phi_1 \end{bmatrix}$ and $R_2=\begin{bmatrix} cos\phi_2&-sin\phi_2\\sin\phi_2&cos\phi_2 \end{bmatrix}$
Step 2. Evaluate the product of matrices:
$R_1R_2=\begin{bmatrix} cos\phi_1&-sin\phi_1\\sin\phi_1&cos\phi_1 \end{bmatrix}\begin{bmatrix} cos\phi_2&-sin\phi_2\\sin\phi_2&cos\phi_2 \end{bmatrix}=\begin{bmatrix} cos\phi_1cos\phi_2-sin\phi_1sin\phi_2&-cos\phi_1sin\phi_2-sin\phi_1cos\phi_2\\sin\phi_1cos\phi_2+cos\phi_1sin\phi_2&-sin\phi_1sin\phi_2+cos\phi_1cos\phi_2 \end{bmatrix}$
Step 3. Use the Addition and Subtract Formulas for Sine and Cosine to simplify the entries of the above matrix, we get:
$R_1R_2=\begin{bmatrix} cos(\phi_1+\phi_2)&-sin(\phi_1+\phi_2)\\sin(\phi_1+\phi_2)&cos(\phi_1+\phi_2) \end{bmatrix}$
Step 4. Compare the above results with the definition of the rotation matrix $R$, we can conclude that $R_1R_2$ represents a rotation through an angle $\phi_1+\phi_2$
