$\begin{array}{l}
\sin \left( {x - \dfrac{\pi }{3}} \right)\cos \left( {\dfrac{{9\pi }}{4} - x} \right) - \sin \left( {\dfrac{{5\pi }}{4} - x} \right)\cos \left( {\dfrac{{5\pi }}{3} + x} \right)\\
= \sin \left( {x - \dfrac{\pi }{3}} \right)\cos \left( {2\pi + \dfrac{\pi }{4} - x} \right) - \sin \left( {\pi + \dfrac{\pi }{4} - x} \right)\cos \left( {2\pi - \dfrac{\pi }{3} + x} \right)\\
= \sin \left( {x - \dfrac{\pi }{3}} \right)\cos \left( {\dfrac{\pi }{4} - x} \right) + \sin \left( {\dfrac{\pi }{4} - x} \right)\cos \left( {x - \dfrac{\pi }{3}} \right)\\
= \sin \left( {x - \dfrac{\pi }{3} + \dfrac{\pi }{4} - x} \right) = - \sin \dfrac{\pi }{{12}}
\end{array}$
