[Math] Matrix for reflection about the line $y = \tan (\theta) \, x$

Question

How would I show that a reflection about the line $y = \tan (\theta) \, x$ is the following?

\begin{pmatrix} \cos 2\theta & \sin 2\theta \\ \sin 2\theta & -\cos 2\theta \end{pmatrix}

Answer 1

Show that$$\begin{bmatrix}\cos(2\theta)&\sin(2\theta)\\ \sin(2\theta)&-\cos(2\theta)\end{bmatrix}.\begin{bmatrix}\cos\theta\\\sin\theta\end{bmatrix}=\begin{bmatrix}\cos\theta\\\sin\theta\end{bmatrix}$$and that$$\begin{bmatrix}\cos(2\theta)&\sin(2\theta)\\ \sin(2\theta)&-\cos(2\theta)\end{bmatrix}.\begin{bmatrix}-\sin\theta\\\cos\theta\end{bmatrix}=\begin{bmatrix}\sin\theta\\-\cos\theta\end{bmatrix}=-\begin{bmatrix}-\sin\theta\\\cos\theta\end{bmatrix}.$$Besides, note that $\left[\begin{smallmatrix}\cos\theta\\\sin\theta\end{smallmatrix}\right]$ belongs to the line $y=\tan(\theta)x$ and that $\left[\begin{smallmatrix}\cos\theta\\\sin\theta\end{smallmatrix}\right]$ and $\left[\begin{smallmatrix}-\sin\theta\\\cos\theta\end{smallmatrix}\right]$ are orthogonal.