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Consider the equation $\cos (7y) = 1$ , then the smallest positive solutions are $y = \frac { 2\pi}{7}, \frac {4 \pi}{7}, \ldots, \frac {14 \pi}{7}$

By Chebyshev's polynomial, $\cos (7y) = 64 \cos^7 (y) - 112 \cos^5 (y) + 56 \cos^3 (y) - 7 \cos (y) = 1$

Let $x = \cos (y)$ , substitution to the above equation followed by division of $(x-1) (\ne 0)$ gives

$(8 x^3 + 4 x^2 - 4 x - 1)^2 = 0$

Which has roots $\pm \cos \left ( \frac {2\pi}{7} \right ), \pm \cos \left ( \frac {4\pi}{7} \right ), \pm \cos \left ( \frac {6\pi}{7} \right )$

This is due to the identity $\cos (\theta) = - \cos (\pi - \theta)$

Let $w = x^2$ then the equation below have roots $\cos^2 \left ( \frac {\pi}{7} \right ), \cos^2 \left ( \frac {2\pi}{7} \right ), \cos^2 \left ( \frac {3\pi}{7} \right )$

$64w^3 - 112w^2 + 56w - 7 = 0$

Now let $W = \frac {1}{w}$ , then multiply both sides by $W^3$ giving

$7W^3 - 56W^2 + 112W - 64 = 0$ to have roots $\sec^2 \left ( \frac {\pi}{7} \right ), \sec^2 \left ( \frac {2\pi}{7} \right ), \sec^2 \left ( \frac {3\pi}{7} \right )$

By Vieta's formula, $\sec^2 \left ( \frac {\pi}{7} \right ) + \sec^2 \left ( \frac {2\pi}{7} \right ) + \sec^2 \left ( \frac {3\pi}{7} \right ) = -\frac {-56}{7} = 8$

Apply the trigonometric identity $\tan^2 (A) + 1 = \sec^2 (A)$ , we should obtain $\displaystyle \sum_{n=1}^3 \tan^2 \left ( \frac {n \pi}{7} \right ) = 5$

By an analogous argument, to evaluate the second summation, let $z = 1 - w$ , the equation below have roots $\sin^2 \left ( \frac {\pi}{7} \right ), \sin^2 \left ( \frac {2\pi}{7} \right ), \sin^2 \left ( \frac {3\pi}{7} \right )$

$64(1-z)^3 - 112(1-z)^2 + 56(1-z) - 7 = 0$ , which simplifies to $64z^3 - 80z^2 + 24z - 1 = 0$ , and let $Z = \frac {1}{z}$ , then multiply both sides by $Z^3$ giving

$Z^3 - 24Z^2 + 80Z - 64 = 0$ to have roots $\csc^2 \left ( \frac {\pi}{7} \right ), \csc^2 \left ( \frac {2\pi}{7} \right ), \csc^2 \left ( \frac {3\pi}{7} \right )$

By Vieta's formula, $\csc^2 \left ( \frac {\pi}{7} \right ) + \csc^2 \left ( \frac {2\pi}{7} \right ) + \csc^2 \left ( \frac {3\pi}{7} \right ) = -\frac {-24}{1} = 24$

Apply the trigonometric identity $\cot^2 (A) + 1 = \csc^2 (A)$ , thus the second summation is equals to $21$ .

The answer is $5 \times 21 = \boxed {105}$