Prove that A $\in M_3(\mathbb{R})$ is similar to $\begin{bmatrix} \cos(x) & -\sin(x) & 0 \\ \sin(x) & \cos(x) & 0 \\ 0 & 0 & \epsilon \end{bmatrix}$

Question

Show that every orthogonal matrix A $\in M_3(\mathbb{R})$ is similar to a matrix $$\begin{bmatrix} cos(x) & -sin(x) & 0 \\\\ sin(x) & cos(x) & 0 \\\\ 0 & 0 & y\end{bmatrix}$$ where $y$ equals either -1 or 1.

We can show that the eigenvalues of the matrix for some angle $x$ is $e^{ix}, e^{-ix}$ and $y$. The corresponding eigenvectors are: $$\begin{bmatrix} i \\\\ 1 \\\\ 0 \end{bmatrix} \begin{bmatrix} -i \\\\ 1 \\\\ 0 \end{bmatrix} \begin{bmatrix} 0 \\\\ 0 \\\\ 1 \end{bmatrix}$$

Now we rewrite the eigenvectors so that $X_1 = U_1+iU_2, X_2 = U_1-iU_2$ and $X_3 = U_3$. Where $X_1, X_2$ and $X_3$ are the eigenvectors. Now we can show that:

$$AU_1=cos(x)U_1 -sin(x)U_2$$ and $$AU_2 = sin(x)U_1 + cos(x)U_2.$$

I have used the hints give in the textbook that I use, but I don't understand the last step and how this shows that the matrices are similar.