Andrei Lazanu proved the double angle formulae illustrated in the diagram:

The diagram starts from a right angled triangle, of sides 2t and 2 and where consequently tanq = t. In this triangle, a line making an angle q with the hypotenuse is drawn. This way, an isosceles triangle is formed, and 2q is the angle exterior to this isosceles triangle. Let the sides DA and DB of this isosceles triangle be x units. Then the length of DC must be 2-x units. Using Pythagoras Theorem for triangle ADC we find x.
x2=(2t)2+(2-x)2.
Hence x=1+t2 and so the length of side DC is 2-(1+t2)=1-t2.

The formulae for the sine, cosine and tangent of 2q in terms of t, where t=tanq, follow directly from the ratios of the sides of the right angled triangle ADC and we get
tan2q = 2t
1-t2
 ,    sin2q = 2t
1+t2
 ,    cos2q = 1-t2
1+t2

Rots and Refs provides a proof of the translation formula .

Finally, to find the combined transformation you have to multiply the transformation matricies to get (using the double angle formulae to simplify):


T = æ
ç
è
cos 2(f-q)
-sin2(f-q)
sin2(f-q
cos2(f-q)
ö
÷
ø
Which is a rotation. If the angles are equal the matrix would be:

T = æ
ç
è
1
0
0
1
ö
÷
ø
So the two reflections combined would leave all points unchanged.