\(E=cos\left(x-\dfrac{\Omega}{3}\right)\cdot cos\left(x+\dfrac{\Omega}{4}\right)+cos\left(x+\dfrac{\Omega}{6}\right)\cdot cos\left(x+\dfrac{3}{4}\Omega\right)\)
\(=\dfrac{1}{2}\left[cos\left(x-\dfrac{\Omega}{3}+x+\dfrac{\Omega}{4}\right)+cos\left(x-\dfrac{\Omega}{3}-x-\dfrac{\Omega}{4}\right)\right]+\dfrac{1}{2}\cdot\left[cos\left(x+\dfrac{\Omega}{6}+x+\dfrac{3}{4}\Omega\right)+cos\left(x+\dfrac{\Omega}{6}-x-\dfrac{3}{4}\Omega\right)\right]\)
\(=\dfrac{1}{2}\cdot\left[cos\left(2x-\dfrac{\Omega}{12}\right)+cos\left(-\dfrac{7}{12}\Omega\right)\right]+\dfrac{1}{2}\left[cos\left(2x+\dfrac{11}{12}\Omega\right)+cos\left(-\dfrac{7}{12}\Omega\right)\right]\)
\(=\dfrac{1}{2}\left[cos\left(2x-\dfrac{\Omega}{12}\right)+cos\left(2x+\dfrac{11}{12}\Omega\right)+cos\left(-\dfrac{7}{12}\Omega\right)+cos\left(-\dfrac{7}{12}\Omega\right)\right]\)
\(=\dfrac{1}{2}\left[cos\left(2x+\dfrac{11}{12}\Omega-\Omega\right)+cos\left(2x+\dfrac{11}{12}\Omega\right)+\dfrac{-\sqrt{6}+\sqrt{2}}{2}\right]\)
\(=\dfrac{1}{2}\left[cos\left[\Omega-\left(2x+\dfrac{11}{12}\Omega\right)\right]+cos\left(2x+\dfrac{11}{12}\Omega\right)+\dfrac{-\sqrt{6}+\sqrt{2}}{2}\right]\)
\(=\dfrac{1}{2}\left[-cos\left(2x+\dfrac{11}{12}\Omega\right)+cos\left(2x+\dfrac{11}{12}\Omega\right)\right]+\dfrac{-\sqrt{6}+\sqrt{2}}{4}=\dfrac{-\sqrt{6}+\sqrt{2}}{4}\)
