Solve it in Friday 13th - part 1

Geometry Level 5

cos(2π13)+cos(6π13)+cos(8π13) \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 αβγ\dfrac{\sqrt{\alpha}-\beta}{\gamma} for positive integers α,β,γ\alpha,\beta,\gamma with where α\alpha doesn't have squared factors and gcd(β,γ)=1\text{gcd}(\beta,\gamma)=1, find α+β+γ\alpha+\beta+\gamma.

See Part 2 and Part 3.

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