I was taught that for the product and sum of two trig function (any function), its period is the smallest common multiple of the two periods.
But I discovered that if the two trig functions have the same angular frequency, then the fundamental frequency of their product is twice either of frequency. This is because they can be made into a double angle trig function. For example, $x(t)=cos(\frac{\pi}{2}t)cos(\frac{\pi}{2}t)$. It is just $\frac{1}{2}+\frac{1}{2}cos(\pi t)$. So the fundamental frequency of the product is actual twice pi over two, which is pi. This is contradicting the rule that I was taught.
But what if the two trig functions have different frequencies. Then you cannot use the double angle identity. So you have to use the smallest common multiple rule. For example $x(t)=cos(2\pi t)sin(\pi t)$. The smallest common multiple of their periods is 2. So the fundamental frequency of their product is pi.
Why are there these two cases? Any more cases?