[tex]\sin \dfrac{11\pi}{12}\cos \dfrac{23\pi}{12} = \sin\Big(\pi - \dfrac{\pi}{12}\Big)\cos\Big(2\pi- \dfrac{\pi}{12}\Big) = \\ \\ = \sin\dfrac{\pi}{12}\cos\dfrac{\pi}{12} = \dfrac{2\sin\dfrac{\pi}{12}\cos\dfrac{\pi}{12}}{2}=\dfrac{\sin \Big(2\cdot \dfrac{\pi}{12}\Big)}{2} =\\ \\ = \dfrac{\sin \dfrac{\pi}{6}}{2} = \dfrac{\dfrac{1}{2}}{2}=\boxed{\dfrac{1}{4}}[/tex]
