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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