A difficult and deep identity of Ramanujan

Abstract:

In his first letter to Hardy, Ramanujan included about 100 formulas. Hardy said of four of these  ``[They] ... are on a different level and obviously both difficult and deep. ... [they] defeated me completely. I had never seen anything in the least like them before. A single look at them is enough to show that they could only be written down by a mathematician of the highest class. They must be true because, if they were not true, no one would have had the imagination to invent them.''

We will give an almost complete proof of one of these identities.

No background is required (beyond induction and roots of unity)!