How do I solve this proof ?

Question

From the definition of a periodic function (f(x) = f(x+p)) prove that the period of f(x) = cos(2x) is π.

Answer 1

We want to prove that p = π for f(x) = cos(2x). We have:

f(x + π) = cos(2(x + π)) = cos(2x + 2π) = cos(2x)cos(2π) - sin(2x)sin(2π) = cos(2x)*1 - sin(2x) * 0 = cos(2x) = f(x)

Therefore, f(x) = f(x + π), and the period is π