Simplify:
`cos theta[{:(costheta,sintheta),(-sintheta,costheta):}]+sintheta[{:(sin theta ,-costheta),(costheta, sintheta):}]`

Explanation

To simplify the expression

\[ \cos \theta \begin{pmatrix} \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{pmatrix} + \sin \theta \begin{pmatrix} \sin \theta & -\cos \theta \\ \cos \theta & \sin \theta \end{pmatrix}, \]

we will follow these steps:

### Step 1: Distribute \(\cos \theta\) and \(\sin \theta\) into their respective matrices.

\[ \cos \theta \begin{pmatrix} \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{pmatrix} = \begin{pmatrix} \cos^2 \theta & \cos \theta \sin \theta \\ -\cos \theta \sin \theta & \cos^2 \theta \end{pmatrix} \]

\[ \sin \theta \begin{pmatrix} \sin \theta & -\cos \theta \\ \cos \theta & \sin \theta \end{pmatrix} = \begin{pmatrix} \sin^2 \theta & -\sin \theta \cos \theta \\ \sin \theta \cos \theta & \sin^2 \theta \end{pmatrix} \]

### Step 2: Add the two resulting matrices.

\[ \begin{pmatrix} \cos^2 \theta & \cos \theta \sin \theta \\ -\cos \theta \sin \theta & \cos^2 \theta \end{pmatrix} + \begin{pmatrix} \sin^2 \theta & -\sin \theta \cos \theta \\ \sin \theta \cos \theta & \sin^2 \theta \end{pmatrix} \]

Adding the corresponding elements gives:

\[ \begin{pmatrix} \cos^2 \theta + \sin^2 \theta & \cos \theta \sin \theta - \sin \theta \cos \theta \\ -\cos \theta \sin \theta + \sin \theta \cos \theta & \cos^2 \theta + \sin^2 \theta \end{pmatrix} \]

### Step 3: Simplify the elements.

1. The first element: \(\cos^2 \theta + \sin^2 \theta = 1\) (using the Pythagorean identity).
2. The second element: \(\cos \theta \sin \theta - \sin \theta \cos \theta = 0\).
3. The third element: \(-\cos \theta \sin \theta + \sin \theta \cos \theta = 0\).
4. The fourth element: \(\cos^2 \theta + \sin^2 \theta = 1\).

Thus, we have:

\[ \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \]

### Final Result:
The simplified expression is the identity matrix:

\[ I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \]