formula">: <em>obscurum per obscurius</em>!  
A different tack might be to start with <img src="pix/sceq42.png" alt="sin in terms of e">, a definition of sine,
and call the "e^(ix)" terms "t", 
then cos(sin) involves terms like <img src="pix/fmeq17.png" alt="cos(sin) in e">,
which is one (convoluted) way to define Bessel functions. 
Or perhaps most forthright, start with the formula for Jn(B) given above (the integral), and say "we want cos(sin) expanded as a sum of cosines, and we
define Jn to be the nth c