You might try to regularize the sum,
$$S(\alpha)=\sum_{n=1}^\infty \frac{ \cos(\alpha n) \sin(n+1) }{n}=
-\tfrac{1}{4} i \left[e^{-i} \ln \left(1-e^{i (\alpha-1)}\right)+e^{-i} \ln \left(1-e^{-i (\alpha+1)}\right)-e^{ i} \ln \left(1-e^{-i (\alpha-1)}\right)-e^{ i}\ln  \left(1-e^{i (\alpha+1)}\right)\right]$$
$$=\text{Re}\, \biggl(\tfrac{1}{2} i e^i \ln \left[2 e^i (\cos 1-\cos \alpha)\right]\biggr),\;\;\alpha\neq\pm 1\;\;\text{mod}\;(2\pi).$$

Here is a plot of $S(\alpha)$.

<IMG SRC="https://i.sstatic.net/11K3l.png" WIDTH="300"/>