Reading about OFDM and it states that orthogonal frequencies can exist and not interfere with one another because of the fact that a sinusoidal wave multiplied by another sinusoid of a different harmonic is 0.

How can you demonstrate mathematically that a sinusoidal wave multiplied by another sinusoid of a different harmonic is 0? Unfortunately the book does show a mathematical expression.

 

By doing the math.

First, however, you need to realize that when they say the result is zero, they mean that the average value goes to zero over a sufficiently long period of time. This is because the average value is exactly zero over any time period that is an integral number of periods of both waveforms.

Consider

x = sin(mwt)
y = sin(nwt)

Now consider

z = x*y

What is the average value of any period T that is a multiple of the periods of both x and y?