Ahaan's trigonometric progression

Geometry Level 4

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

This problem is posed by Ahaan Rungta.

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