Alex,

At your page:

http://www.cut-the-knot.org/pythagoras/cosine.shtml

You mention "I'll be extremely curious to learn of any proof of the Cosine Rule completely independent of the Pythagorean Theorem."

I have one. It is quite simple, and quite possibly someone has already done this.

We have three equations resulting from drawing the three altitudes in a triangle ABC:

1) c = bcosA + acosB
2) b = acosC + ccosA
3) a = bcosC + ccosB

From equation 2) we have cosA = (b - acosC)/c
From equation 3) we have cosB = (a - bcosC)/c

Substituting for cosA and cosB into equation 1) gives the Law of Cosines.