Double Trouble

Calculus Level 5

\[S_n = \left(1-\tan^4\frac{\pi}{2^3}\right) \left(1-\tan^4\frac{\pi}{2^4}\right) \left(1-\tan^4\frac{\pi}{2^5}\right) \ldots \left(1-\tan^4\frac{\pi}{2^n}\right)\]

For \(S_n\) as defined above, evaluate \(\displaystyle30.28 \lim_{n\to\infty}\frac{\pi^3}{S_n}\) to the nearest Integer.