I am hopeful that I am on the right track for what is being asked to verify in exercise 8.7. I'm not 100% sure of the accuracy though. See the insert.
Thanks Tim

If the transformation of the Riemann sphere given by those equations is supposed to be a 90° rotation around the line joining the points +1 and -1 , then these two points should remain invariant, but I am stuck allready because when z=1, t becomes 0.
Evidently there is something essential I am missing here.
Where am I wrong?

Perhaps it is enough to show that z = (i-t)/(i+t) maps the real numbers to the unit circle. Well, we know the modulus of z is 1 when t is real (the moduli of the upper and lower bits are the same) and we are told that the number line is mapped to a circle under a bilinear mapping. We know that the circle passes through i, 1, and -i (by substituting 1, 0 and -1 for t ) so therefore it must be the unit circle.
Well, its supposed to be an easy problem!

Tony
This exercise is not marked as simple in my book. It has the symbol "needs a bit of thought" next to it.
Have you looked at my solution?
Here is the link to it, 8.7b http://www.roadtoreality.info/viewtopic.php?f=19&t=286