period of the sum and the product of $\sin(ax)$ and $\cos(bx)$

Question

Take the function f(x)=sin(ax)cos(bx) , with a,b>0 . We know that the fundamental period of sin(ax) is p= 2π/a and the fundamental period of cos(bx) is q=2π/b. Suppose there are positive integer n and m such that np=mq=r, with n/m reduced to its lowest term; then r is a period of f(x) but not always the shortest : for f(x)=sinx⋅cos(3x) we have r = 2π , but the fundamental period is π. Now, consider the sum instead of the product of sin(ax) and cos(bx).If g(x)=sin(ax)+cos(bx) , r should always be the fundamental period of g(x) ? And if so, how to prove it?