Euler’s squares

On account of our fascination with the geometry of origami (albeit not well-endowed in mathematical capacity) we discovered for ourselves shortly after our father had taught us trigonometry that,

We had earlier shown the origami proof for that. But it was only a little later while drifting away from one of those trigonometric identities that you routinely faced in those annoying college exams we stumbled upon a beautiful relationship that was apparently first discovered by the great Leonhard Euler. This is a relationship parallel to the above one:

The proof for this, like the origami proof for the above, can be achieved from a self-evident construction of Euler — what the Hindus of yore would have called an upapatti or mathematicians today term “wordless” proof. It is illustrated below but I add several words for the benefit of the non-geometrically oriented reader.

1) Draw square ABCD and triplicate it so that the three squares share a side.
2) Draw diagonals AC and CG of first two squares and use them to draw square ACGF and duplicate it.
3) Draw : from the construction it is apparent that
4) From the construction it is clear that
5) We thus see:

This relationship is one of a class of strange trigonometric relationships that interestingly bring in the meru-średhī (called in western literature as Fibonacci sequence):

, the meru-średhī; thus for , we get .
This leads us to a formula for based on the odd terms of meru-średhī starting from :

Shown below is the convergence of the above series to : We reach an accuracy of 6 decimal places for .