Solve it in Friday 13th - part 1

Geometry Level 5

\[ \large \cos\left(\dfrac{2\pi}{13}\right)+\cos\left(\dfrac{6\pi}{13}\right)+\cos\left(\dfrac{8\pi}{13}\right)\]

If the trigonometric expression above can be expressed as \(\dfrac{\sqrt{\alpha}-\beta}{\gamma}\) for positive integers \(\alpha,\beta,\gamma\) with where \(\alpha\) doesn't have squared factors and \(\text{gcd}(\beta,\gamma)=1\), find \(\alpha+\beta+\gamma\).

See Part 2 and Part 3.

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