# Jacobi series for Bessel's function

Using Jacobi series, prove the following

$${J_0}^2+{J_1}^2+{J_2}^2+\cdots=1$$

My trial:

$$\cos(x\sin θ)=J_0+2J_2\cos(2θ) +2J_4\cos(4θ)+\cdots$$

$$\sin(x\sinθ)=2\big(J_1\sin(θ) +J_3\sin(3θ)+J_5\sin(5θ)+\cdots\big)$$

Squaring both equations and adding them is what I thought we are supposed to do but in this is not enough to solve this equality. How do i proceed from here

• The ()'s are oddly placed, please fix that. – emacs drives me nuts 2 days ago
• Weird equations, there is no $x$ on the RHS. – vonbrand 2 days ago