$\cos\dfrac{\pi}{7}.\cos\dfrac{4\pi}{7}.\cos\dfrac{5\pi}{7}$
$=\dfrac{1}{2}(\cos\dfrac{5\pi}{7}+\cos\dfrac{3\pi}{7}).\cos\dfrac{5\pi}{7}$
$=\dfrac{1}{2}\cos\dfrac{3\pi}{7}+\dfrac{1}{2}\cos^2\dfrac{5\pi}{7}$
$=\dfrac{1}{2}\cos\dfrac{3\pi}{7}+\dfrac{1}{2}(\dfrac{1}{2}+\dfrac{1}{2}\cos\dfrac{10\pi}{7})$
$=\dfrac{1}{2}\cos\dfrac{3\pi}{7}+\dfrac{1}{4}\cos\dfrac{10\pi}{7}+\dfrac{1}{4}$
