Fourier series help!?

When you multiply two sinusoidal waves, you'll get the sum and the difference, as per trig identities:

sin(x+y)=sin x cos y + cos x sin y

sin(x-y)=sin x cos y - cos x sin y

If we add both together, we get 1/2 sin(x+y) + 1/2 sin(x-y)= sin x cos y

The frequencies will be

5+3=8

5-3=2

The amplitudes will both be 0.5.

sin 3 cos 5= 1/2 sin(3+5) + 1/2 sin(3-5)

=0.5 sin(8t) + 0.5 sin(-2t)

But sin(-2t)=-sin(2t), so:

0.5 sin 8t - 0.5 sin 2t.