Just solve for the sum in the addition formula (31) of [Neumann 1867](//zbmath.org/01.0139.02), [p. 65](//archive.org/details/theoriederbesse00neumgoog/page/n79/mode/1up) (also in Watson, [p. 128](//archive.org/details/treatiseontheory00watsuoft/page/128/mode/1up)):
$$
J_0\left(2r\sqrt{\frac{1-\cos\theta}2}\right)
= J_0(r)^2 + 2\sum_{n=1}^\infty J_n(r)^2\cos(n\theta).
$$