Use the diagram to prove the double angle formula, where

t = \tan θ

\tan 2 θ = \frac{2 t}{1 - t^{2}}, \sin 2 θ = \frac{2 t}{1 + t^{2}}, \cos 2 θ = \frac{1 - t^{2}}{1 + t^{2}}

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' = p \cos 2 θ + q \sin 2 θ, q' = p \sin 2 θ - q \cos 2 θ (1)

where

m = \tan θ

. Hence establish another proof that the matrix

T_{2} = (\begin{matrix} \cos 2 θ & \sin 2 θ \sin 2 θ & - \cos 2 θ \end{matrix})

gives a reflection in the line

y = x \tan θ

. The point

P'' = (p'', q'')

is the image of the point

P'

after reflection in the line

y = x \tan Φ

. Apply the transformation

T'_{2} = (\begin{matrix} \cos 2 Φ & \sin 2 Φ \sin 2 Φ & - \cos 2 Φ \end{matrix})

to the point

P' = (p', q')

to find the coordinates of the point

P''

in terms of

p, q, θ

and

Φ

. 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.

θ = Φ