[tex]\displaystyle \cos\left(\dfrac{7\pi}{12}\right) =\cos\left(\dfrac{3\pi+4\pi}{12}\right) =\\\\=\cos\left(\dfrac{3\pi}{12}+\dfrac{4\pi}{12}\right) = \cos\left(\dfrac{\pi}{4}+\dfrac{\pi}{3}\right) =\\ \\ =\cos\left(\dfrac{\pi}{4}\right)\cos \left(\dfrac{\pi}{3}\right)-\sin\left(\dfrac{\pi}{4}\right)\sin \left(\dfrac{\pi}{3}\right)=\\ \\ = \dfrac{\sqrt{2}}{2}\cdot \dfrac{1}{2}-\dfrac{\sqrt{2}}{2}\cdot \dfrac{\sqrt{3}}{2} = \dfrac{\sqrt{2}-\sqrt{6}}{4}[/tex]
