ok let me ask the question this way.
why (besides the fact that it is true) is
\[\cos\left(\frac{\pi}{5}\right)\] the same as the solution to
\[4x^2-2x-1=0\]

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maybe this is just not that interesting, but it is bugging me

I don't get what you mean. \(\cos(\frac{\pi}{5})\) is not a root for \(4x^2−2x−1\).

Oh but it is. One does not lightly challenge the Great Satellite.

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