Alternative Derivation of Recurrence Relation for Bessel Functions of the First Kind How can the recurrence relation
$J^{'}_n(x) = \frac{1}{2} [J_{n-1}(x) - J_{n+1}(x)]$
be derived directly from the following?
$J_n(x) = \frac{1}{\pi} \int_0^\pi \cos(n\theta - x \sin\theta) \text{d} \theta$

Hint: $J'_n(x)={1\over\pi}\int_0^{\pi}\sin(\theta)\sin(n\theta-x\sin\theta)d\theta$
Hint: use $\sin x\sin y ={1\over 2}(\cos(x-y)-\cos(x+y))$
$={1\over \pi}\int_0^{\pi}{1\over 2}(\cos(-\theta+n\theta-x\sin\theta)-\cos(\theta+n\theta-x\sin\theta))d\theta$
