Find the axis of reflection

Question

I have to determine the axis of reflection of the composition of a rotation and a reflection, y show that the order of composition matters.
So I multiply the matrices that represent each isometry, with angles $\alpha$ and $\beta$ not necessarily the same, and I got: $$Re\alpha \circ Ro\beta= \left( \begin{matrix} cos 2\alpha & sin 2\alpha \\ sin 2\alpha & -cos2\alpha & \\ \end{matrix} \right) \left( \begin{matrix} cos \beta & -sin \beta \\ sin \beta & cos \beta& \\ \end{matrix} \right)$$ $$= \left( \begin{matrix} cos \beta \;cos2\alpha+sin2\alpha \;sin\beta & -sin \beta \;cos2\alpha+sin2\alpha \;cos\beta\\ sin 2\alpha \;cos\beta-cos2\alpha \;sin\beta & -sin2\alpha \;sin\beta-cos2\alpha \;sin\beta& \\ \end{matrix} \right)$$ Well, then I wanted to write this last matrix as a reflection, $\left( \begin{matrix} cos \gamma & sin \gamma\\ sin \gamma & -cos \gamma \\ \end{matrix} \right)$, where $\gamma$ would give me the axis, but since $\alpha$ and $\beta$ might not be the same, I don't think I can use the same trigonometric identities.

Answer 1

Use the trigonometric identities $$\sin(A\pm B)=\sin A\cos B \pm\cos A\sin B$$ $$\cos(A\pm B)=\cos A\cos B \mp\sin A\sin B$$ For more identities see http://en.wikipedia.org/wiki/List_of_trigonometric_identities

Answer 2

By using trigonometric identities for the sum of two angles, you should verify that this last matrix is actually a reflection matrix corresponding to (half) the angle $2\alpha-\beta$. In other words, the first component of the matrix is $\cos(2\alpha-\beta)$, and so on.