Duplicate of 1823
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)?