Use the diagram to prove the double angle formula, where t=tanq:
tan2q = 2t
1-t2
 ,    sin2q = 2t
1+t2
 ,    cos2q = 1-t2
1+t2

Tan
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 ' =pcos2q+ qsin2q, q ' =psin2q- qcos2q    (1)
where m=tanq. Hence establish another proof that the matrix
T2= æ
ç
è
cos2q
sin2q
sin2q
-cos2q
 ö
÷
ø
gives a reflection in the line y=xtanq. The point P ' ' =(p ' ' ,q ' ' ) is the image of the point P ' after reflection in the line y=xtanf. Apply the transformation
T ' 2= æ
ç
è
cos2f
sin2f
sin2f
-cos2f
 ö
÷
ø
to the point P ' =(p ' ,q ' ) to find the coordinates of the point P ' ' in terms of p, q, q and f. 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. q = f)?