A geometry problem by Harry Jones

$\begin{aligned} S & = \frac{\sin^2{\frac{2\pi }{7}}}{\sin^2{\frac{\pi }{7}}}+\frac{\sin^2{\frac{4\pi }{7}}}{\sin^2{\frac{2\pi }{7}}}+\frac{\color{#3D99F6} \sin^2{\frac{\pi }{7}}}{\sin^2{\frac{4\pi }{7}}} \\ & = \frac{\sin^2{\frac{2\pi }{7}}}{\sin^2{\frac{\pi }{7}}}+\frac{\sin^2{\frac{4\pi }{7}}}{\sin^2{\frac{2\pi }{7}}}+\frac{\color{#3D99F6} \left(-\sin \frac{8\pi }{7}\right)^2}{\sin^2{\frac{4\pi }{7}}} \\ & = \sum_{n=0}^2 \frac{\sin^2\frac{2^{n+1}\pi}7}{\sin^2 \frac{2^n\pi}7} \\ & = \sum_{n=0}^2 \frac{4\sin^2\frac{2^n\pi}7 \cos^2 \frac {2^n\pi}7} {\sin^2 \frac{2^n\pi}7} \\ & = \sum_{n=0}^2 4 \cos^2 \frac {2^n\pi}7 & \small \color{#3D99F6} \text{Note that: } \cos^2 x = \frac 12(\cos 2x +1) \\ & = 2 \sum_{n=0}^2 \left(\cos \frac {2^{n+1}\pi}7 +1\right) \\ & = 2 \left({\color{#3D99F6}\cos \frac {2\pi}7 + \cos \frac {4\pi}7 + \cos \frac {8\pi}7} \right) + 6 & \small \color{#3D99F6} \text{Note that: } \cos (\pi - x) = - \cos x \\ & = {\color{#3D99F6}-} 2 \left({\color{#3D99F6}\cos \frac {5\pi}7 + \cos \frac {3\pi}7 + \cos \frac {\pi}7} \right) + 6 & \small \color{#3D99F6} \text{See note: } \sum_{k=0}^{n-1} \cos \left( \frac {2k+1}{2n+1}\pi \right) = \frac 12 \\ & = - 2 { \color{#3D99F6} \left(\frac 12\right)} + 6 \\ & = \boxed{5} \end{aligned}$

Note:

$\small \begin{aligned} S & = \sum_{k=0}^{n-1} \cos \left( \frac {2k+1}{2n+1}\pi \right) \\ & = \Re \left \{ \sum_{k=0}^{n-1} e^{\frac {2k+1}{2n+1}\pi i} \right \} \\ & = \Re \left \{e^{\frac {\pi i}{2n+1}} \sum_{k=0}^{n-1} e^{\frac {2k\pi i}{2n+1}} \right \} \\ & = \Re \left \{e^{\frac {\pi i}{2n+1}} \left(\frac {1-e^{\frac {2n\pi i}{2n+1}}}{1-e^{\frac {2\pi i}{2n+1}}}\right) \right \} \\ & = \Re \left \{ \frac {e^{\frac {\pi i}{2n+1}}-e^{\pi i}}{1-e^{\frac {2\pi i}{2n+1}}} \right \} \\ & = \Re \left \{ \frac {e^{\frac {\pi i}{2n+1}} + 1}{\left(1+e^{\frac {\pi i}{2n+1}} \right) \left(1-e^{\frac {\pi i}{2n+1}} \right)} \right \} \\ & = \Re \left \{ \frac 1{1-e^{\frac {\pi i}{2n+1}}} \right \} \\ & = \Re \left \{ \frac 1{1-\cos \frac {\pi i}{2n+1} - i\sin \frac {\pi i}{2n+1}} \right \} \\ & = \Re \left \{ \frac {1-\cos \frac {\pi i}{2n+1} + i\sin \frac {\pi i}{2n+1}}{\left(1-\cos \frac {\pi i}{2n+1}\right)^2 + \sin^2 \frac {\pi i}{2n+1}} \right \} \\ & = \Re \left \{ \frac {1-\cos \frac {\pi i}{2n+1} + i\sin \frac {\pi i}{2n+1}}{2-2\cos \frac {\pi i}{2n+1}} \right \} \\ & = \frac 12 \end{aligned}$