Evaluate :
`[{:(,cos 45^@, sin 30^@),(,sqrt2 cos 0^@, sin 0^@):}] [{:(,sin 45^@, cos 90^@),(,sin 90^@, cot 45^@):}]`

Answer

To evaluate the given expression, we will first substitute the trigonometric values into the matrices and then perform the matrix multiplication step by step.

### Step 1: Substitute the trigonometric values

We will substitute the trigonometric values into the matrices:

1. **Matrix A**:
- \( \cos 45^\circ = \frac{1}{\sqrt{2}} \)
- \( \sin 30^\circ = \frac{1}{2} \)
- \( \sqrt{2} \cos 0^\circ = \sqrt{2} \cdot 1 = \sqrt{2} \)
- \( \sin 0^\circ = 0 \)

Thus, Matrix A becomes:
\[ A = \begin{pmatrix} \frac{1}{\sqrt{2}} & \frac{1}{2} \\ \sqrt{2} & 0 \end{pmatrix} \]

2. **Matrix B**:
- \( \sin 45^\circ = \frac{1}{\sqrt{2}} \)
- \( \cos 90^\circ = 0 \)
- \( \sin 90^\circ = 1 \)
- \( \cot 45^\circ = 1 \)

Thus, Matrix B becomes:
\[ B = \begin{pmatrix} \frac{1}{\sqrt{2}} & 0 \\ 1 & 1 \end{pmatrix} \]

### Step 2: Set up the matrix multiplication

Now we will multiply Matrix A and Matrix B:
\[ AB = \begin{pmatrix} \frac{1}{\sqrt{2}} & \frac{1}{2} \\ \sqrt{2} & 0 \end{pmatrix} \begin{pmatrix} \frac{1}{\sqrt{2}} & 0 \\ 1 & 1 \end{pmatrix} \]

### Step 3: Perform the matrix multiplication

To find the product \( AB \), we will calculate each element of the resulting matrix:

1. **First row, first column**:
\[ \left(\frac{1}{\sqrt{2}} \cdot \frac{1}{\sqrt{2}}\right) + \left(\frac{1}{2} \cdot 1\right) = \frac{1}{2} + \frac{1}{2} = 1 \]

2. **First row, second column**:
\[ \left(\frac{1}{\sqrt{2}} \cdot 0\right) + \left(\frac{1}{2} \cdot 1\right) = 0 + \frac{1}{2} = \frac{1}{2} \]

3. **Second row, first column**:
\[ \left(\sqrt{2} \cdot \frac{1}{\sqrt{2}}\right) + \left(0 \cdot 1\right) = 1 + 0 = 1 \]

4. **Second row, second column**:
\[ \left(\sqrt{2} \cdot 0\right) + \left(0 \cdot 1\right) = 0 + 0 = 0 \]

### Step 4: Combine the results

Thus, the resulting matrix \( AB \) is:
\[ AB = \begin{pmatrix} 1 & \frac{1}{2} \\ 1 & 0 \end{pmatrix} \]

### Final Result

The evaluated result of the matrix multiplication is:
\[ \begin{pmatrix} 1 & \frac{1}{2} \\ 1 & 0 \end{pmatrix} \]