Anger three-term recurrence

The following formula holds: $$\textbf{J}_{\nu-1}(z)+\textbf{J}_{\nu+1}(z)=\dfrac{2\nu}{z}\textbf{J}_{\nu}(z)-\dfrac{2}{\pi z}\sin(\pi \nu),$$ where $\textbf{J}_{\nu}$ denote the Anger function, $\pi$ denotes pi, and $\sin$ denotes sine.