Problem
2 [2 points] Let \( \phi, \theta \in \mathbb{R} \). Use a linear transformation to prove that \[ \cos (\phi+\theta)=\cos \phi \cos \theta-\sin \phi \sin \theta \] and \[ \sin (\phi+\theta)=\sin \phi \cos \theta+\cos \phi \sin \theta \]
Solution
Step 1 ：\[\begin{bmatrix} \cos\phi & -\sin\phi \\ \sin\phi & \cos\phi \end{bmatrix} \begin{bmatrix} \cos\theta \\ \sin\theta \end{bmatrix} = \begin{bmatrix} \cos\phi\cos\theta - \sin\phi\sin\theta \\ \sin\phi\cos\theta + \cos\phi\sin\theta \end{bmatrix}\]
Step 2 ：\[\cos (\phi+\theta) = \cos \phi \cos \theta - \sin \phi \sin \theta\]
Step 3 ：\[\sin (\phi+\theta) = \sin \phi \cos \theta + \cos \phi \sin \theta\]
