Simplify the expression: $\cos 0 + \cos \dfrac{\pi}{7} + \cos \dfrac{2 \pi}{7} + \cos \dfrac{3 \pi}{7} + \cos \dfrac{4 \pi}{7} + \cos \dfrac{5 \pi}{7} + \cos \dfrac{6 \pi}{7}$
$\cos 0 + \cos \dfrac{\pi}{7} + \cos \dfrac{2 \pi}{7} + \cos \dfrac{3 \pi}{7} + \cos \dfrac{4 \pi}{7} + \cos \dfrac{5 \pi}{7} + \cos \dfrac{6 \pi}{7}$
$= 1 + \left(\cos \dfrac{6 \pi}{7} + \cos \dfrac{\pi}{7}\right) + \left(\cos \dfrac{5 \pi}{7} + \cos \dfrac{2 \pi}{7}\right) + \left(\cos \dfrac{4 \pi}{7} + \cos \dfrac{3 \pi}{7}\right)$
$= 1 + 2 \cos \left(\dfrac{\dfrac{6 \pi}{7} + \dfrac{\pi}{7}}{2}\right) \cos \left(\dfrac{\dfrac{6 \pi}{7} - \dfrac{\pi}{7}}{2}\right)$
$\hspace{1.5cm} + 2 \cos \left(\dfrac{\dfrac{5 \pi}{7} + \dfrac{2\pi}{7}}{2}\right) \cos \left(\dfrac{\dfrac{5 \pi}{7} - \dfrac{2\pi}{7}}{2}\right)$
$\hspace{2.5cm} + 2 \cos \left(\dfrac{\dfrac{4\pi}{7} + \dfrac{3\pi}{7}}{2}\right) \cos \left(\dfrac{\dfrac{4\pi}{7} - \dfrac{3\pi}{7}}{2}\right)$
$= 1 + 2 \cos \dfrac{\pi}{2} \cos \dfrac{5 \pi}{14} + 2 \cos \dfrac{\pi}{2} \cos \dfrac{3 \pi}{14} + 2 \cos \dfrac{\pi}{2} \cos \dfrac{\pi}{14}$
$= 1 + 2 \times 0 \times \cos \dfrac{5 \pi}{14} + 2 \times 0 \times \cos \dfrac{3\pi}{14} + 2 \times 0 \times \cos \dfrac{\pi}{14}$
$= 1$