$\begin{array}{l} A = \cos \left( {x - \dfrac{\pi }{3}} \right)\cos \left( {x + \dfrac{\pi }{4}} \right) + \cos \left( {x + \dfrac{\pi }{6}} \right)\cos \left( {x + \dfrac{{3\pi }}{4}} \right)\\ A = \dfrac{1}{2}\left[ {\cos \left( {2x - \dfrac{\pi }{{12}}} \right) + \cos \dfrac{{ - 7\pi }}{{12}}} \right] + \dfrac{1}{2}\left[ {\cos \left( {2x + \dfrac{{11\pi }}{{12}}} \right) + \cos \dfrac{{ - 7\pi }}{{12}}} \right]\\ A = \dfrac{1}{2}\cos \left( {2x - \dfrac{\pi }{{12}}} \right) + \dfrac{1}{2}\cos \dfrac{{7\pi }}{{12}} + \dfrac{1}{2}\cos \left( {\pi  + 2x - \dfrac{\pi }{{12}}} \right) + \dfrac{1}{2}\cos \dfrac{{7\pi }}{{12}}\\ A = \dfrac{1}{2}\cos \left( {2x - \dfrac{\pi }{{12}}} \right) - \dfrac{1}{2}\cos \left( {2x - \dfrac{\pi }{{12}}} \right) + \cos \dfrac{{7\pi }}{{12}} = \cos \dfrac{{7\pi }}{{12}} \end{array}$

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

$\begin{array}{l} B = {\sin ^2}x + {\sin ^2}\left( {{{60}^o} + x} \right) + {\sin ^2}\left( {{{60}^o} - x} \right)\\ B = \dfrac{{1 - \cos 2x}}{2} + \dfrac{{1 - \cos \left( {{{120}^o} + 2x} \right)}}{2} + \dfrac{{1 - \cos \left( {{{120}^o} - 2x} \right)}}{2}\\ B = \dfrac{{1 - \cos 2x}}{2} + \dfrac{{1 - \cos \left( {{{180}^o} + 2x - {{60}^o}} \right)}}{2} + \dfrac{{1 - \cos \left( {{{180}^o} - {{60}^o} - 2x} \right)}}{2}\\ B = \dfrac{{1 - \cos 2x + 1 + \cos \left( {2x - {{60}^o}} \right) + 1 + \cos \left( {{{60}^o} + 2x} \right)}}{2}\\ B = \dfrac{{2 - \cos 2x + 2\cos {{60}^o}\cos 2x}}{2} = 1 \end{array}$