[tex]\begin{array}{rcl}a &=& \cos(90-43)^{\circ}\cos(90-77)^{\circ} - \cos77^{\circ}\cos43^{\circ} \\ \\ &=& \sin43^{\circ}\sin77^{\circ} - \cos 77^{\circ}\cos 43^{\circ} \\ \\ &=& -\Big( \cos 77^{\circ}\cos 43^{\circ} - \sin43^{\circ}\sin77^{\circ}\Big) \\ \\ &=& - \cos(77+43)^{\circ} \\ \\ &=&-\cos120^{\circ} \\ \\ &=& -\cos\Big(90-(-30)\Big)^{\circ} \\ \\ &=& -\sin(-30)^{\circ} \\ \\ &=& \sin30^{\circ} \\ \\ &=& \boxed{\dfrac{1}{2}} \in \mathbb{Q} \end{array}[/tex]
