$\newcommand{\bbx}[1]{\,\bbox[15px,border:1px groove navy]{\displaystyle{#1}}\,}
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$\ds{\LARGE a)}$
With the
Abel-Plana Formula:
\begin{align}
\sum_{n = 1}^{\infty}{\pars{-1}^{n}\sin\pars{n} \over n} & =
-1 + \sum_{n = 0}^{\infty}\pars{-1}^{n}\,\mrm{sinc}\pars{n} =
-1 + {1 \over 2}\,\mrm{sinc}\pars{0} = -1 + {1 \over 2} = \bbx{-\,{1 \over 2}}
\end{align}
$\ds{\LARGE b)}$
\begin{align}
\sum_{n = 1}^{\infty}{\pars{-1}^{n}\sin\pars{n} \over n} & =
\Im\sum_{n = 1}^{\infty}{\pars{-1}^{n}\expo{\ic n} \over n} =
\Im\sum_{n = 1}^{\infty}{\pars{-\expo{\ic}}^{n} \over n} =
-\,\Im\ln\pars{1 + \expo{\ic}}
\\[5mm] & =
-\,\Im\ln\pars{1 + \cos\pars{1} + \sin\pars{1}\,\ic} =
-\arctan\pars{\sin\pars{1} \over 1 + \cos\pars{1}}
\\[5mm] & =
-\arctan\pars{2\sin\pars{1/2}\cos\pars{1/2} \over 2\cos^{2}\pars{1/2}} =
-\arctan\pars{\tan\pars{1 \over 2}} = \bbx{-\,{1 \over 2}}
\end{align}