Chapter 12.23, Problem 22MP

In the $\cos (x\sin \theta )$ series of Problem 20, let $\theta =0,$ and then let $\theta =\pi /2,$ and add the results to show that (recall Problem 13.2) ${\displaystyle \underset{n=-\infty }{\overset{\infty }{\sum }}{J}_{4n}}(x)=\frac{1}{2}(1+\cos x)$.