$\begin{array}{l}
\sin \left[ {\left( {ax + \dfrac{\pi }{2}} \right) + \dfrac{\pi }{2}} \right]\\
= \sin \left( {ax + \dfrac{\pi }{2}} \right)\cos \dfrac{\pi }{2} + \cos \left( {ax + \dfrac{\pi }{2}} \right)\sin \dfrac{\pi }{2}\\
= \sin \left( {ax + \dfrac{\pi }{2}} \right).0 + \cos \left( {ax + \dfrac{\pi }{2}} \right).1\\
= 0 + \cos \left( {ax + \dfrac{\pi }{2}} \right)\\
= \cos \left( {ax + \dfrac{\pi }{2}} \right)
\end{array}$
