<s>[INST] Rotation Matrix and Triple Angle Formulas? 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?
 [/INST] You may use these identities
$\cos x \cos y - \sin x \sin y = \cos(x+y)$
$\sin x \cos y + \cos x \sin y = \sin(x+y)$
and use $\theta $ and $2\theta$ in place of $x$ and $y$. For $R_{n\theta}$, try proving it by induction, assuming $(R_{\theta})^n = R_{n\theta}$ to be true and find $(R_{\theta})^{n+1}$ 
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