Evaluation of Sum of $ \sum_{n=1}^{\infty}\frac{\sin (n)}{n}$.
Rewrite what you have as:
\begin{align} 2S =& i \ln(1-e^i) -i \ln(1 - e^{-i} ) \\ =& i \ln \left (\frac{1-e^i}{1-e^{-i}} \right ) \\ =&i \ln \left ( -e^{i} \frac{1- e^{-i}}{1-e^{-i}} \right ) \\ =& i \big ( \ln ( -e^{i}) \big) \\ =& i \big ( \ln ( e^{-i \pi} e ^{i} \big ) \\ =& i \big ( \ln ( e^{ i(1 -\pi)} \big ) \\ =& i \big( i ( 1- \pi) \big)\\ =& \pi -1 \\ \end{align}
by choosing the right branch.