If `P(x)=[cosxsinx-sinxcosx]` , then show that `P(x)\ P(y)=P(x+y)=P(y)\ P(x)` .

Explanation

To show that \( P(x) P(y) = P(x+y) = P(y) P(x) \), where \( P(x) = \begin{pmatrix} \cos x & \sin x \\ -\sin x & \cos x \end{pmatrix} \), we will follow these steps:

### Step 1: Calculate \( P(x) P(y) \)

We start by multiplying the matrices \( P(x) \) and \( P(y) \):

\[
P(x) = \begin{pmatrix} \cos x & \sin x \\ -\sin x & \cos x \end{pmatrix}, \quad P(y) = \begin{pmatrix} \cos y & \sin y \\ -\sin y & \cos y \end{pmatrix}
...