# Splendid Series

Geometry Level 4

$a_n=\cos\left(\frac{\pi}{2n+1}\right) \cos\left(\frac{2\pi}{2n+1}\right) \cos\left(\frac{3\pi}{2n+1}\right)\cdots \cos\left(\frac{(2n+1)\pi}{2n+1}\right)$

Let $$a_n$$ be defined as above for non-negative integer $$n$$ and $$S=a_0+a_1+a_2+\cdots$$. What is $$\dfrac{1}{1+S}=\boxed{?}$$

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