If `A=[[costheta,isintheta],[isintheta,costheta]],` then prove by principal of mathematical induction that `A^n=[[cosntheta, i sinn theta],[isin n theta, cos n theta]]` for all n`in` `NN`.

Explanation

To prove that \( A^n = \begin{pmatrix} \cos(n\theta) & i \sin(n\theta) \\ i \sin(n\theta) & \cos(n\theta) \end{pmatrix} \) for all \( n \in \mathbb{N} \), we will use the principle of mathematical induction.

### Step 1: Base Case
We start with \( n = 1 \).

\[ A^1 = A = \begin{pmatrix} \cos(\theta) & i \sin(\theta) \\ i \sin(\theta) & \cos(\theta) \end{pmatrix} \]
...