Relationship between spherical Bessel y and cosine

Theorem

The following formula holds for non-negative integers $n$: $$y_n(z)=(-1)^{n+1}z^n \left( \dfrac{1}{z} \dfrac{d}{dz} \right)^n \left( \dfrac{\cos z}{z} \right),$$ where $y_n$ denotes the spherical Bessel function of the second kind and $\cos$ denotes the cosine function.