Evalutate:
`[{:(,2cos 60^@, -2sin 30^@),(,-tan 45^@,cos 0^@):}] [{:(,cot 45^@, cosec 30^@),(,sec 60^@, sin 90^@):}]`

Answer

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

### Step 1: Substitute the trigonometric values into the matrices.

The given expression is:

\[ \begin{pmatrix} 2\cos 60^\circ & -2\sin 30^\circ \\ -\tan 45^\circ & \cos 0^\circ \end{pmatrix} \begin{pmatrix} \cot 45^\circ & \csc 30^\circ \\ \sec 60^\circ & \sin 90^\circ \end{pmatrix} \]

Now, we will substitute the trigonometric values:

- \(\cos 60^\circ = \frac{1}{2}\)
- \(\sin 30^\circ = \frac{1}{2}\)
- \(\tan 45^\circ = 1\)
- \(\cos 0^\circ = 1\)
- \(\cot 45^\circ = 1\)
- \(\csc 30^\circ = 2\)
- \(\sec 60^\circ = 2\)
- \(\sin 90^\circ = 1\)

Substituting these values, we get:

\[ \begin{pmatrix} 2 \cdot \frac{1}{2} & -2 \cdot \frac{1}{2} \\ -1 & 1 \end{pmatrix} \begin{pmatrix} 1 & 2 \\ 2 & 1 \end{pmatrix} \]

This simplifies to:

\[ \begin{pmatrix} 1 & -1 \\ -1 & 1 \end{pmatrix} \begin{pmatrix} 1 & 2 \\ 2 & 1 \end{pmatrix} \]

### Step 2: Perform matrix multiplication.

Now we will multiply the two matrices:

\[ \begin{pmatrix} 1 & -1 \\ -1 & 1 \end{pmatrix} \begin{pmatrix} 1 & 2 \\ 2 & 1 \end{pmatrix} \]

To find the element in the first row and first column:

\[ 1 \cdot 1 + (-1) \cdot 2 = 1 - 2 = -1 \]

To find the element in the first row and second column:

\[ 1 \cdot 2 + (-1) \cdot 1 = 2 - 1 = 1 \]

To find the element in the second row and first column:

\[ -1 \cdot 1 + 1 \cdot 2 = -1 + 2 = 1 \]

To find the element in the second row and second column:

\[ -1 \cdot 2 + 1 \cdot 1 = -2 + 1 = -1 \]

Putting it all together, we get:

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

### Final Answer:

The evaluated matrix is:

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