Bessel function of order 1

Prove that $\displaystyle J_1(x)=\frac{1}{\pi}\int_0^\pi\cos(\theta-x\sin\theta)d\theta$ by showing that the right-hand side satisfies Bessel's equation of order 1 and that the derivative has the value $\displaystyle J_1'(0)$ when $\displaystyle x=0$. Explain why this constitutes a proof.