Average electric field over a spherical surface

[JD_PM]

Homework Statement

I was working out problem 4, chapter 3 of Introduction to Electrodynamics by Griffiths:

a) Show that the average electric field over a spherical surface, due to charges outside the sphere, is the same as the field at the centre.

b) What is the average due to charges inside the sphere?

The Attempt at a Solution

I know the solution, but there are a few aspects I do not see:

Here you have the image Griffiths makes reference at:

Why $z^2 + r^2 - R^2 = z - Rcos \theta$ (law of cosines developement)?

 

[TSny]

Why $z^2 + r^2 - R^2 = z - Rcos \theta$ (law of cosines developement)?

This equality is not correct. Instead, note the following substitution