Define $R_{\theta}:\mathbb{R}^2 \rightarrow \mathbb{R}^2$ as the rotation matrix by angle $\theta$, where
$$R_{\theta} = \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}$$
Observe that
$$ (R_\theta)^2 = \begin{pmatrix} \cos^2\theta-\sin^2\theta & -2\sin\theta\cos\theta \\ 2\sin\theta\cos\theta & \cos^2\theta-\sin^2\theta \end{pmatrix} = \begin{pmatrix} \cos2\theta & -\sin2\theta \\ \sin2\theta & \cos2\theta \end{pmatrix}=R_{2\theta} $$
This all makes sense of course since if you rotate a vector by $\theta$ twice, the net result should be a rotation by $2\theta$. The algebra of it all can be verified with the double angle formulas.
However, how do you prove that
$$ (R_\theta)^3 = R_{3\theta} $$
or perhaps that
$$ (R_\theta)^n = R_{n\theta} $$
Are there triple angle formulas that can be used to make the algebra work? n-tuple angle formulas?