Relationship between spherical Bessel j and sine

Theorem

The following formula holds for non-negative integers $n$: $$j_n(z)=(-1)^nz^n \left( \dfrac{1}{z} \dfrac{\mathrm{d}}{\mathrm{d}z} \right)^n \left( \dfrac{\sin z}{z} \right),$$ where $j_n$ denotes the spherical Bessel function of the first kind and $\sin$ denotes the sine function.