## The Powers of the Matrix with Cosine and Sine Functions

## Problem 567

Prove the following identity for any positive integer $n$.

\[\begin{bmatrix}

\cos \theta & -\sin \theta\\

\sin \theta& \cos \theta

\end{bmatrix}^n=\begin{bmatrix}

\cos n\theta & -\sin n\theta\\

\sin n\theta& \cos n\theta

\end{bmatrix}.\]