=> $\int \large\frac{\sin (x - \alpha + \alpha)}{\sin (x - \alpha)}$$dx$
=> $\int \large\frac{\sin (x - \alpha) \cos \alpha}{\sin (x - \alpha)}+ \int \large\frac{\sin \cos (x - \alpha)}{\sin (x - \alpha)}$
=> $ \int \cos \alpha .dx + \int \sin \alpha \cot (x- \alpha)dx$
$x \cos \alpha + \sin \alpha \log | \sin (x - \alpha)| +c$
Hence a is the correct answer.