Squared cosines

$\begin{aligned} S & = \cos^2 10^\circ + \cos^2 50^\circ + \cos^2 70^\circ \\ & = \cos^2 \frac \pi{18} + \cos^2 \frac {5\pi}{18} + \cos^2 \frac {7\pi}{18} \\ & = \frac 12 \left(\cos \frac {\pi}{9} + 1 \right) + \frac 12 \left(\cos \frac {5\pi}{9} + 1 \right) + \frac 12 \left(\cos \frac {7\pi}{9} + 1 \right) \\ & = \frac 12 \left(\cos \frac {\pi}{9} + \cos \frac {5\pi}{9} + \cos \frac {7\pi}{9} \right) + \frac 32 \\ & = \frac 12 \left({\color{#3D99F6}\cos \frac {\pi}{9} + \cos \frac {3\pi}{9} + \cos \frac {5\pi}{9} + \cos \frac {7\pi}{9}} - \cos \frac {3\pi}{9} \right) + \frac 32 & \small \color{#3D99F6} \text{See Note: } \sum_{k=0}^{n-1} \cos \left(\frac {2k+1}{2n+1}\pi \right) = \frac 12 \\ & = \frac 12 \left({\color{#3D99F6}\frac 12} - \frac 12 \right) + \frac 32 \\ & = \frac 32 = \boxed{1.5} \end{aligned}$

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Note:

$\begin{aligned} S & = \sum_{k=0}^{n-1} \cos \left(\frac {2k+1}{2n+1}\pi \right) \\ & = \sum_{k=0}^{n-1} \Re \left \{ e^{\frac {2k+1}{2n+ 1} \pi i} \right \} & \small \color{#3D99F6} \text{where } \Re \{ z \} \text{ is the real part of complex number }z \\ & = \Re \left \{\sum_{k=0}^{n-1} e^{\frac {2k+1}{2n+1}\pi i} \right \} \\ & = \Re \left \{e^{\frac {\pi i}{2n+1}} \sum_{k=0}^{n-1} e^{\frac {2k \pi i}{2n+1}} \right \} \\ & = \Re \left \{e^{\frac {\pi i}{2n+1}} \left( \frac {1 - e^{\frac {2n \pi i}{2n+1}}}{1 - e^{\frac {2\pi i}{2n+1}}} \right) \right \} \\ & = \Re \left \{\frac {e^{\frac {\pi i}{2n+1}} + 1}{1 - e^{\frac {2\pi i}{2n+1}}} \right \} \\ & = \Re \left \{\frac 1{1 - e^{\frac {\pi i}{2n+1}}} \right \} \\ & = \Re \left \{\frac 1{1 - \cos {\frac \pi{2n+1}} - i \sin {\frac \pi{2n+1}}} \right \} \\ & = \Re \left \{\frac {1 - \cos {\frac \pi{2n+1}} + i \sin {\frac \pi{2n+1}}}{\left(1 - \cos {\frac \pi{2n+1}}\right)^2 + \sin^2 {\frac \pi{2n+1}}} \right \} \\ & = \Re \left \{\frac {1 - \cos {\frac \pi{2n+1}} + i \sin {\frac \pi{2n+1}}}{2 - 2\cos {\frac \pi{2n+1}}} \right \} \\ & = \boxed{\dfrac 12} \end{aligned}$