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\(\dfrac{{\sin \alpha + \sin \beta \cos \left( {\alpha + \beta } \right)}}{{\cos \alpha - \sin \beta \sin \left( {\alpha + \beta } \right)}}\)\( = \dfrac{{\sin \alpha + \dfrac{1}{2}\left[ {\sin \left( {\alpha + 2\beta } \right) + \sin \left( { - \alpha } \right)} \right]}}{{\cos \alpha + \dfrac{1}{2}\left[ {\cos \left( {\alpha + 2\beta } \right) - \cos \left( { - \alpha } \right)} \right]}}\) \( = \dfrac{{\sin \alpha + \dfrac{1}{2}\left[ {\sin \left( {\alpha + 2\beta } \right) - \sin \alpha } \right]}}{{\cos \alpha + \dfrac{1}{2}\left[ {\cos \left( {\alpha + 2\beta } \right) - \cos \alpha } \right]}}\)\( = \dfrac{{\sin \alpha + \dfrac{1}{2}\sin \left( {\alpha + 2\beta } \right) - \dfrac{1}{2}\sin \alpha }}{{\cos \alpha + \dfrac{1}{2}\cos \left( {\alpha + 2\beta } \right) - \dfrac{1}{2}\cos \alpha }}\)\( = \dfrac{{\dfrac{1}{2}\sin \alpha + \dfrac{1}{2}\sin \left( {\alpha + 2\beta } \right)}}{{\dfrac{1}{2}\cos \alpha + \dfrac{1}{2}\cos \left( {\alpha + 2\beta } \right)}}\) \( = \dfrac{{\sin \left( {\alpha + 2\beta } \right) + \sin \alpha }}{{\cos \left( {\alpha + 2\beta } \right) + \cos \alpha }}\) \( = \dfrac{{2\sin \dfrac{{\left( {\alpha + 2\beta + \alpha } \right)}}{2}\cos \dfrac{{\left( {\alpha + 2\beta - \alpha } \right)}}{2}}}{{2\cos \dfrac{{\left( {\alpha + 2\beta + \alpha } \right)}}{2}\cos \dfrac{{\left( {\alpha + 2\beta - \alpha } \right)}}{2}}}\)\( = \dfrac{{2\sin \left( {\alpha + \beta } \right)\cos \beta }}{{2\cos \left( {\alpha + \beta } \right)\cos \beta }} = \tan \left( {\alpha + \beta } \right)\)