15/$\begin{array}{l} \sin a + 1 = \sin a + \sin \dfrac{\pi }{2} = 2\sin \left( {\dfrac{a}{2} - \dfrac{\pi }{4}} \right)\cos \left( {\dfrac{a}{2} + \dfrac{\pi }{4}} \right)\\ \to A \end{array}$
16/$\begin{array}{l} \sin a = \sqrt {\dfrac{1}{3}} \Rightarrow \cos a = \sqrt {1 - {{\sin }^2}a} = \sqrt {\dfrac{2}{3}} (0 < a < \dfrac{\pi }{2})\\ \cos \left( {a + \dfrac{\pi }{3}} \right) = \cos a.\cos \dfrac{\pi }{3} - \sin a.\sin \dfrac{\pi }{3} = \sqrt {\dfrac{2}{3}} .\dfrac{1}{2} - \sqrt {\dfrac{1}{3}} .\dfrac{{\sqrt 3 }}{2} = \dfrac{{\sqrt 6 - 3}}{6}\\ \to A \end{array}$
