OK, so I lied about the maths
But this is all there is, honest

If the carrier is the signal we want to shift and the modulator is a sinewave
of the frequency we want to shift by, and we can phase shift the carrier by 90
degrees somehow, then:



The phase shift applied to the carrier is transfered through to the modualtion
sidebands, and a phase shift of 180 degrees is very like inverting (negating) it.

and a sinewave phase shifted by 90 degrees is just a cosine wave, so: