Lời giải chi tiết:

Ta có:

\(\begin{array}{l}
\dfrac{{\sin \alpha + \sin \beta c{\rm{os}}\left( {\alpha + \beta } \right)}}{{\cos \alpha - \sin \beta \sin \left( {\alpha + \beta } \right)}}\\
= \dfrac{{\sin \alpha + \dfrac{1}{2}\left[ {\sin \left( {\alpha + 2\beta } \right) - {\rm{sin}}\left( \alpha \right)} \right]}}{{\cos \alpha - \dfrac{1}{2}\left[ {\cos \left( {\alpha + 2\beta } \right) - \cos \left( \alpha \right)} \right]}}\\
= \dfrac{{\sin \left( {\alpha + 2\beta } \right){\rm{ + sin}}\left( \alpha \right)}}{{\cos \left( {\alpha + 2\beta } \right) + \cos \left( \alpha \right)}}\\
= \dfrac{{2\sin \left( {\alpha + \beta } \right)\cos \left( \beta \right)}}{{2\cos \left( {\alpha + \beta } \right)\cos \left( \beta \right)}}\\
= \tan \left( {\alpha + \beta } \right)
\end{array}\)

Chọn A.