How can we derive the Rodrigue's formula for $j_n(z)$ from its integral representation?

Question

the integral representation of the spherical Bessel function of first kind $$j_{n}\left(z\right)=\frac{z^{n}}{2^{n+1}n!}\int_{0}^{\pi}\cos\left(z\cos\theta\right)(\sin\theta)^{2n+1}\,d\theta.\tag{1}$$ Starting from (1), can we arrive at the formula $$j_n(z)=(-1)^{n}z^n\left(\frac{1}{z}\frac{d}{dz}\right)^n\left(\frac{\sin z}{z}\right)?$$

Answer 1

For $n=0$ you can integrate directly, then apply $\frac1z \frac{d}{dz}$ to the integral for $j_n(z)z^{-n}$ and integrate by parts