a) \(\sin(a + b) + \sin\left (  \frac{\pi }{2} - a \right )\sin(-b) \)

\(= \sin a\cos b + \cos a\sin b - \cos a\sin b \)

\(= \sin a\cos b\)

b) \(\cos( \frac{\pi }{4} + a)\cos(\frac{\pi }{4}- a) + \frac{1 }{2}\sin^2a\)

\( =\frac{1 }{2}\cos\left [ \frac{\pi }{4}+a+\frac{\pi}{4} -a\right ]+\frac{1}{2}\cos\left [ \left ( \frac{\pi }{4} +a\right ) -\left ( \frac{\pi}{4}-a \right )\right ]+\frac{1}{2}\left ( \frac{1-\cos 2a}{2} \right )\)

\( =\frac{1}{2}\cos 2a +  \frac{1}{4}(1 - \cos 2a) \)

\(= \frac{1+\cos 2a}{4 }= \frac{1 }{2}\cos^2 a\)

c) \(\cos( \frac{\pi}{2} - a)\sin( \frac{\pi}{2} - b) - \sin(a - b) \)

\(= \sin a\cos b - \sin a\cos b + \sin b\cos a\)

\(= \cos a\sin b\)