$\begin{array}{l} A = \sin \left( {\dfrac{\pi }{4} + x} \right) - \cos \left( {\dfrac{\pi }{4} + x} \right)\\ A = \sqrt 2 \left( {\dfrac{{\sqrt 2 }}{2}\sin \left( {\dfrac{\pi }{4} + x} \right) - \dfrac{{\sqrt 2 }}{2}\cos \left( {\dfrac{\pi }{4} + x} \right)} \right)\\ A = \sqrt 2 \left( {\cos \dfrac{\pi }{4}\sin \left( {\dfrac{\pi }{4} + x} \right) - \sin \dfrac{\pi }{4}\cos \left( {\dfrac{\pi }{4} + x} \right)} \right)\\ A = \sqrt 2 \left( {\sin \left( {\dfrac{\pi }{4} + x - \dfrac{\pi }{4}} \right)} \right) = \sqrt 2 \sin x \end{array}$
b/
$\begin{array}{l} \cos \left( {\dfrac{\pi }{6} - x} \right) + \sin \left( {\dfrac{\pi }{3} + x} \right)\\ A = \cos \dfrac{\pi }{6}\cos x + \sin \dfrac{\pi }{6}\sin x + \sin \dfrac{\pi }{3}\cos x + \sin x\cos \dfrac{\pi }{3}\\ A = \dfrac{{\sqrt 3 }}{2}\cos x + \dfrac{1}{2}\sin x + \dfrac{{\sqrt 3 }}{2}\cos x + \dfrac{1}{2}\sin x\\ A = \sqrt 3 \cos x + 1\sin x\\ A = 2.\left( {\dfrac{{\sqrt 3 }}{2}\cos x + \dfrac{1}{2}\sin x} \right)\\ A = 2\sin \left( {\dfrac{\pi }{6} + x} \right) \end{array}$
c/
$\begin{array}{l}
C = {\sin ^2}x + \cos \left( {\dfrac{\pi }{3} - x} \right)\cos \left( {\dfrac{\pi }{3} + x} \right)\\
C = \dfrac{{1 - \cos 2x}}{2} + \dfrac{1}{2}\left[ {\cos 2x + \cos \dfrac{{2\pi }}{3}} \right]\\
C = \dfrac{{1 - \cos 2x + \cos 2x + \cos \dfrac{{2\pi }}{3}}}{2} = \dfrac{3}{4}
\end{array}$
