Explain the meaning of $z_{j}$ in the Weierstrass canonical product formula

$$f(z)=f(0) \exp \left[\frac{f^{\prime}(0)}{f(0)} z\right] \prod_{j=1}^{\infty}\left\{\left(1-\frac{z}{z_{j}}\right) e^{\frac{z}{z_{j}}}\right\}$$

Show that

$$\frac{\sin (\pi z)}{\pi z}=\prod_{n=1}^{\infty}\left(1-\frac{z^{2}}{n^{2}}\right)$$

Deduce that

$$\pi \cot (\pi z)=\frac{1}{z}+2 \sum_{n=1}^{\infty} \frac{z}{z^{2}-n^{2}}$$