`cos theta [(cos theta, sin theta),(-sin theta, cos theta)]+sin theta[(sin theta, -cos theta),(cos theta, sin theta)]` is equal to

Options

A. `[(1,0),(0,1)]`
B. `[(0,1),(1,0)]`
C. `[(0,0),(1,1)]`
D. None of these

Answer

To solve the given 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: Multiply \(\cos \theta\) with the first matrix
We will multiply \(\cos \theta\) with each element of the first matrix:

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

### Step 2: Multiply \(\sin \theta\) with the second matrix
Next, we will multiply \(\sin \theta\) with each element of the second matrix:

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

### Step 3: Add the two resulting matrices
Now we will add the two matrices obtained from the previous steps:

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

Adding corresponding elements, we get:

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

### Step 4: Simplify the resulting matrix
Now we can simplify the matrix:

1. The first element simplifies to \(\cos^2 \theta + \sin^2 \theta = 1\).
2. The second element simplifies to \(\cos \theta \sin \theta - \sin \theta \cos \theta = 0\).
3. The third element simplifies to \(-\cos \theta \sin \theta + \sin \theta \cos \theta = 0\).
4. The fourth element simplifies to \(\cos^2 \theta + \sin^2 \theta = 1\).

Thus, the resulting matrix is:

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

### Final Result
The final result of the expression is:

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