Reflect again

Use the diagram to prove the double angle formula, where $t=\tan \theta$: $$\tan 2\theta =\frac{2t}{1-t^2},\sin 2\theta =\frac{2t}{1+t^2},\cos 2\theta =\frac{1-t^2}{1+t^2}$$

Image

The point $P'=(p',q')$ is the image of the point $P=(p,q)$ after reflection in the line $y=mx$. To find $(p',q')$ use the fact that the midpoint of $PP'$ is on the line $y=mx$ and the line segment $PP'$ is perpendicular to the line $y=mx$ and show that $$p^{\prime }=p\cos 2\theta +q\sin 2\theta ,q^{\prime }=p\sin 2\theta -q\cos 2\theta (1)$$ where $m=\tan\theta$. Hence establish another proof that the matrix

$$T_2=\left(\begin{array}{cc}\cos 2\theta & \sin 2\theta \\ \sin 2\theta & -\cos 2\theta \end{array}\right)$$

gives a reflection in the line $y=x\tan\theta$. The point $P''=(p'',q'')$ is the image of the point $P'$ after reflection in the line $y=x\tan\phi$. Apply the transformation $$T_2^{\prime }=\left(\begin{array}{cc}\cos 2\varphi & \sin 2\varphi \\ \sin 2\varphi & -\cos 2\varphi \end{array}\right)$$ to the point $P'=(p',q')$ to find the coordinates of the point $P''$ in terms of $p, q, \theta$ and $\phi$. Hence show that the combination of two reflections in distinct intersecting lines is a rotation about the point of intersection by twice the angle between the two mirror lines. What is the effect of the two reflections if the lines coincide (i.e. $\theta=\phi$)?