One of my favorite resolutions of the Basel problem ($\zeta(2)=\sum_{n=1}^\infty\frac{1}{n^2}=\frac{\pi^2}{6}$) uses an inverse square law for light intensity as an inspiration for a Euclidean geometry proof of the identity
$$ \csc^2 x=\sum_{-\infty~}^{~\infty} \frac{1}{(x-n\pi)^2} \tag{1} $$
The proof can be found here (with a different normalization on $x$).
Can anybody find / think of a Euclidean geometry proof of either of these in the same vein?
$$ \begin{array}{ll} \cot x & \displaystyle = \textrm{p.v.}~ \sum_{-\infty~}^{~\infty} \frac{1}{x-n\pi} \\ & \displaystyle =\frac{1}{x}+\sum_{n=1}^\infty \frac{2x}{x^2-n^2\pi^2}. \end{array} \tag{2} $$
$$ \begin{array}{ll} \displaystyle \frac{\sin x}{x} & \displaystyle = \textrm{p.v.}~ \prod_{n\ne0}\left(1-\frac{x}{n\pi}\right) \\ & \displaystyle = \prod_{n=1}^\infty\left(1-\frac{x^2}{n^2\pi^2}\right). \end{array} \tag{3} $$