How do I prove that solutions to the simple pendulum differential equation are always periodic?

Question

I want to prove that all nontrivial solutions to the second order ODE $\dfrac{d^2x}{dt^2}=-\sin(x)$, for all initial conditions are always periodic functions. The idea I had was to convert it to a system of first order differential equations and calculating the Jacobian, but how can one then conclude the periodicity of the solutions?