Ahaan's trigonometric progression

Geometry Level 4

Let θ\theta be an acute angle which satisfies the equation n=0sin2n+1θ=1.\sum_{n = 0}^{\infty} \sin^{2n + 1} \theta = 1. The value of 2tan8θ 2 \, \tan^8 \theta can be expressed in the form abc a - b \sqrt {c} , where a,b a, b , and c c are positive integers and cc not a multiple of the square of any prime. What is the value of a+b+c a+b+c ?

This problem is posed by Ahaan Rungta.

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