Integral definition of Bessel function is equivalent to power series representation

Question

Let $$J_n(z)=\frac{1}{\pi}\int_0^\pi \cos(z\sin(t)-nt)\mathrm{d}t$$ be the $n$-th Bessel function. I need to prove that this definition is equivalent to $$J_n(z)=\sum_{m=0}^\infty \frac{(-1)^m}{m!(m+n)!}\left(\frac{z}{2}\right)^{n+2m}$$ I have noticed that both satisfy the equation $z^2J_n''(z)+zJ_n'(z)+(z^2-n^2)J_n(z)=0$, but the result does not follow from this alone.