I saw the following two identities and wanted to know whether they can be proved to be equivalent. Can you help with this?
$$ J_{0}(a)=\frac{1}{\pi}\int_{0}^{\pi}\cos(a\sin ϕ)\mathrm dϕ $$ (Reference)
and $$ J_{0}(a)=\frac{1}{2\pi}\int_{0}^{2\pi}\cos(a\cosϕ)\mathrm dϕ $$ where $a=\text{const}$ and $J_{0}$ is a Bessel function of the first kind.
First, remark that due to the evenness of the integrand in the first identity, we may write $$J_0(z)=\frac{1}{\pi}\int_0^\pi\cos(z\sin\phi)\mathrm d\phi=\frac{1}{2\pi}\int_{-\pi}^\pi\cos(z\sin\phi)\mathrm d\phi$$ Now use the change of variable $\theta=\pi/2-\phi$ to get $$J_0(z)=\frac{-1}{2\pi}\int_{3\pi/2}^{-\pi/2}\cos(z\sin(\pi/2-\theta))\mathrm d\theta \\ =\frac{1}{2\pi}\int_{-\pi/2}^{3\pi/2}\cos(z\cos\theta)\mathrm d\theta$$ And because the integrand is $2\pi$ periodic, its integral over any interval of length $2\pi$ is the same, hence $$J_0(z)=\frac{1}{2\pi}\int_{0}^{2\pi}\cos(z\cos\theta)\mathrm d\theta$$