`if A=[{:(costheta,sin theta ),(-sin theta,costheta):}]`, then show that : `A^(2)=[{:(cos2theta,sin2theta),(-sin2theta,cos2theta):}]`

Explanation

To show that \( A^2 = \begin{pmatrix} \cos 2\theta & \sin 2\theta \\ -\sin 2\theta & \cos 2\theta \end{pmatrix} \) for the matrix \( A = \begin{pmatrix} \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{pmatrix} \), we will perform matrix multiplication step by step.

### Step 1: Write down the matrix A
Given:
\[ A = \begin{pmatrix} \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{pmatrix} \]

### Step 2: Multiply A by itself
To find \( A^2 \), we need to calculate \( A \times A \):
\[ A^2 = A \times A = \begin{pmatrix} \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{pmatrix} \times \begin{pmatrix} \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{pmatrix} \]

### Step 3: Calculate the elements of the resulting matrix
We will calculate each element of the resulting matrix \( A^2 \).

1. **Element at (1,1)**:
\[ A_{11} = \cos \theta \cdot \cos \theta + \sin \theta \cdot (-\sin \theta) = \cos^2 \theta - \sin^2 \theta \]

2. **Element at (1,2)**:
\[ A_{12} = \cos \theta \cdot \sin \theta + \sin \theta \cdot \cos \theta = 2 \sin \theta \cos \theta \]

3. **Element at (2,1)**:
\[ A_{21} = -\sin \theta \cdot \cos \theta + \cos \theta \cdot (-\sin \theta) = -\sin \theta \cos \theta - \sin \theta \cos \theta = -2 \sin \theta \cos \theta \]

4. **Element at (2,2)**:
\[ A_{22} = -\sin \theta \cdot \sin \theta + \cos \theta \cdot \cos \theta = \cos^2 \theta - \sin^2 \theta \]

### Step 4: Combine the results into a matrix
Now we can write the resulting matrix \( A^2 \):
\[ A^2 = \begin{pmatrix} \cos^2 \theta - \sin^2 \theta & 2 \sin \theta \cos \theta \\ -2 \sin \theta \cos \theta & \cos^2 \theta - \sin^2 \theta \end{pmatrix} \]

### Step 5: Use trigonometric identities
Using the double angle formulas:
- \( \cos 2\theta = \cos^2 \theta - \sin^2 \theta \)
- \( \sin 2\theta = 2 \sin \theta \cos \theta \)

We can rewrite \( A^2 \):
\[ A^2 = \begin{pmatrix} \cos 2\theta & \sin 2\theta \\ -\sin 2\theta & \cos 2\theta \end{pmatrix} \]

### Conclusion
Thus, we have shown that:
\[ A^2 = \begin{pmatrix} \cos 2\theta & \sin 2\theta \\ -\sin 2\theta & \cos 2\theta \end{pmatrix} \]