Before revision: However, when the input is a sinusoid of the form , using product-to-sum trigonometric identities it can be easily shown that the output is , that is, the output doesn't consist only of sinusoids of same frequency as the input (), but instead also of sinusoids of frequencies and ; furthermore, the fundamental frequency of the input is , which is different than that of the input.
After revision: However, when the input is a sinusoid of the form , using product-to-sum trigonometric identities it can be easily shown that the output is , that is, the output doesn't consist only of sinusoids of same frequency as the input (), but instead also of sinusoids of frequencies and ; furthermore, taking the least common multiple of the fundamental period of the sinusoids of the output, it can be shown the fundamental angular frequency of the output is , which is different than that of the input.