Matrix and trigonometry

$\large \begin{pmatrix} \cos\theta & \sin\theta \\ -\sin\theta & \cos\theta \end{pmatrix}^n = \begin{pmatrix} \cos n\theta & \sin n\theta \\ -\sin n\theta & \cos n\theta \end{pmatrix}$

$\large \begin{pmatrix} \frac{\sqrt{3}+1}{2\sqrt{2}} & \frac{\sqrt{3}-1}{2\sqrt{2}} \\ \frac{1-\sqrt{3}}{2\sqrt{2}} & \frac{1+\sqrt{3}}{2\sqrt{2}} \end{pmatrix}^{2017} = \begin{pmatrix} \cos15^o & \sin15^o \\ -\sin15^o & \cos15^o \end{pmatrix}^{2017} = \begin{pmatrix} \cos(2017\cdot15)^o & \sin(2017\cdot15)^o \\ -\sin(2017\cdot15)^o & \cos(2017\cdot15)^o \end{pmatrix} = \begin{pmatrix} \cos15^o & \sin15^o \\ -\sin15^o & \cos15^o \end{pmatrix}$

$\large \begin{pmatrix} \frac{\sqrt{6}-\sqrt{2}}{4} & \frac{\sqrt{2}+\sqrt{6}}{4} \\ \frac{-\sqrt{2}-\sqrt{6}}{4} & \frac{\sqrt{6}-\sqrt{2}}{4} \end{pmatrix}^{k} = \begin{pmatrix} \cos75^o & \sin75^o \\ -\sin75^o & \cos75^o \end{pmatrix}^{k} = \begin{pmatrix} \cos75k^o & \sin75k^o \\ -\sin75k^o & \cos75k^o \end{pmatrix}$

$\large 75k \equiv 15 \pmod {360}$

$\large \Rightarrow k = 5+24n$ for $\large n =0, 1 , 2 , \cdots$