$\begin{array}{l} C = \sin \left( {x + {{10}^o}} \right)\cos \left( {2x - {{80}^o}} \right) + \sin \left( {x + {{100}^o}} \right)\cos \left( {2x + {{10}^o}} \right)\\ \;\;\; = \sin \left( {x + {{10}^o}} \right)\cos \left( {2x - {{80}^o}} \right) + \sin \left( {{{90}^o} + x + {{10}^o}} \right)\cos \left( {{{90}^o} + 2x - {{80}^o}} \right) = \sin \left( {x + {{10}^o}} \right)\cos \left( {2x - {{80}^o}} \right) + \cos \left( {x + {{10}^o}} \right)\left[ { - \sin \left( {2x - {{80}^o}} \right)} \right]\\ C = \sin \left( {x + {{10}^o}} \right)\cos \left( {2x - {{80}^o}} \right) - \cos \left( {x + {{10}^o}} \right)\sin \left( {2x - {{80}^o}} \right) = \sin \left( {x + {{10}^o} - 2x + {{80}^o}} \right) = \sin \left( {{{90}^o} - x} \right) = \cos x \end{array}$  

d)$\begin{array}{l} F = \sin \left( {x - \dfrac{\pi }{3}} \right)\cos \left( {\dfrac{{9\pi }}{4} - x} \right) - \sin \left( {\dfrac{{5\pi }}{4} - x} \right)\cos \left( {\dfrac{{5\pi }}{3} + x} \right) = \sin \left( {x - \dfrac{\pi }{3}} \right)\cos \left( {\dfrac{\pi }{4} - x} \right) - \sin \left( {\pi  + \dfrac{\pi }{4} - x} \right)\cos \left( {2\pi  - \dfrac{\pi }{3} + x} \right)\\ F = \sin \left( {x - \dfrac{\pi }{3}} \right)\cos \left( {\dfrac{\pi }{4} - x} \right) + \sin \left( {\dfrac{\pi }{4} - x} \right)\cos \left( {x - \dfrac{\pi }{3}} \right) = \sin \left( {x - \dfrac{\pi }{3} + \dfrac{\pi }{4} - x} \right) = \sin \left( {\dfrac{\pi }{4} - \dfrac{\pi }{3}} \right) = \sin \dfrac{\pi }{4}\cos \dfrac{\pi }{3} - \sin \dfrac{\pi }{3}\cos \dfrac{\pi }{4} = \dfrac{{\sqrt 2 }}{2}.\dfrac{1}{2} - \dfrac{{\sqrt 2 }}{2}.\dfrac{{\sqrt 3 }}{2} = \dfrac{{ - \sqrt 6  + \sqrt 2 }}{4} \end{array}$