The Trigonometric Pie

Geometry Level pending

\[\large f ( \theta) =\frac { \sin ^{ 2 }{ (4\theta } )-\sin ^{ 2 }{ (\theta ) } }{ \sin { (\theta )\sin { (5\theta ) } } } \]

Let function \(f(\theta)\) be defined as above. Find \(\displaystyle \prod _{n=0}^8 \left(f (2^{n-1} \theta)-1\right) \), where \(\theta =\dfrac \pi {513} \).

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