Also possibly of interest:
$$ \sum_{n=1}^\infty \left(-\frac{2}{b(n-1/b)(n+1/b)} + \frac{b}{n(n+1)}\right) = \pi \cot(\pi/b)$$
$$ \sum_{n=1}^\infty {\frac {t \left( {t}^{2}{n}^{2}+2\,{n}^{2}+2\,n+1 \right) }{ \left( n+ 1 \right) n \left( {t}^{2}{n}^{2}+1 \right) }} = \pi \coth(\pi/t) $$