Trigo Lovers cannot be short of problems

Geometry Level 4

\[\large \left( \frac12 + \cos\left(\frac\pi{20}\right)\right) \left( \frac12 + \cos\left(\frac{3\pi}{20}\right)\right) \left( \frac12 + \cos\left(\frac{9\pi}{20}\right)\right) \left( \frac12 + \cos\left(\frac{27\pi}{20}\right)\right) = \ ? \]

Give your answer to 4 decimal places.

This is a very very beautiful problem. This is a part of Trigo Plus!.
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