Fastest Way to Cool Boiling Water with Ice and Water

Natural density says numbers below any fixed bound are negligible. Countable additivity says you still cannot draw one uniformly. Here is where the two separate.

Why the tangent inside the sine drops out, two independent proofs of the closed form for every even power, and the cases where both proofs stop working.

Product formulas, Hankel-Parseval and Schwinger parameters solve the lower powers. Here is how far each one gets on the fifth.