I need to prove, knowing the (first order) Bessel function:
$$J_n(x) = \frac{1}{\pi}\int_0^\pi \cos(nt - x\sin(t)) \ dt$$
that the following equality is true:
$$J_{n-1}(x) + J_{n+1}(x) = \frac{2n}{x}J_n$$
After a long time trying to solve this on my own, I opened the correction book. Here is a step I can't understand:
I suspect an integration by parts, since the derivative factor of $\cos(nt - x\sin (t))$, which is $n-x\cos(t)$, appears.
Any idea would be helpful.