DEMODULATION FOR SINUSOIDAL AM
$ y(t) = x(t)cos{w_{c}t} $
the original signal can be recovered by modulating y(t) with the same sinusoidal carrier and applying a lowpass filter to the result.
consider,
$ w(t) = y(t)cos{w_{c}t} $
$ w(t) = x(t)cos^2{w_{c}t} $
using trigonometric identity,
$ cos^2{w_{c}t}=\frac{1}{2}+\frac{1}{2}cos{2w_{c}t} $
$ w(t)=\frac{1}{2}x(t)+\frac{1}{2}x(t)cos{2w_{c}t} $
