Learning About RF Mixers

I follow both SolderSmoke and Ham Radio Work Bench, so I've been avidly watching the SolderSmoke driect conversion reeiver challenge thrown down by Bill Meara. Bill's said something for year's that's intrigued me: you gotta be able to understand how an RF mixer works. That was paraphrased, of course. I haven't had much success along those lines until today.

Looking at the Wikipedia entry for Frequency Mixers, I bludgeoned my way through the diode section, and I finally get it. (OK, I'm a little miffed that I have to take on faith the expression for the current through an ideal diode, but I can let that slide for the moment.) Lo and behold, after a series expansion and a binomial multiplicaiotn, there was the promised expression for the sine of one frequency times the sine of another. Using a high school trig identity, that can be rewritten as a sum of waves one of which is the sum of the two frequencies and one of which is the difference.

That's not all though. Someone pointed out that this is an example of something called prosthaphaeresis, a method for multiplying arbitrary numbers when tables of trig functions are available, but not calculaotors. Apparently, this was used quite frequently in navigation with spherical trig! The Wiki page on prosthaphaeresis also has this spiffy proof-without-words:

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