Follow hints to investigate the matrix which gives a reflection of the plane in the line y=tanx. Show that the combination of two reflections in intersecting lines is a rotation.

Beautiful mathematics. Two 18 year old students gave eight
different proofs of one result then generalised it from the 3 by 1
case to the n by 1 case and proved the general result.

T for Tan

Age 16 to 18 Challenge Level:

Use the diagram to prove the double angle formula, where
$t=\tan \theta$: $$\tan2\theta = {2t\over {1-t^2}}, \quad
\sin2\theta = {2t\over {1+t^2}},\quad \cos2\theta = {{1-t^2}\over
{1+t^2}}$$