prove that function f(x)=sinx +sinax
is cyclic if and only if "a" is rational
It's harder to prove the converse. Suppose that f(x) is periodic. Then will also be periodic, with the same period. But when x=0, f(0)=f''(0)=0. If there is another point, say x=c, at which f(c)=f''(c)=0, and a≠±1, then you should be able to show that sin(c) = sin(ac) = 0. With c≠0, this is only possible if a is rational. But if there is no such point c then f cannot be periodic.
But if then x must be a multiple of π. Therefore both c and ac are multiples of π, say c=nπ and ac=mπ. Then a=m/n, which is rational.