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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

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  • $\begingroup$ The ()'s are oddly placed, please fix that. $\endgroup$ – emacs drives me nuts Feb 14 at 8:24
  • $\begingroup$ Weird equations, there is no $x$ on the RHS. $\endgroup$ – vonbrand Feb 14 at 13:21

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