Sine infinite product ($\sin z=z{\prod }_{n\ge 1}(1-z^2/({\pi }^2{n}^2))$)

The paired zero factors converge locally uniformly because $\sum n^{-2}$ converges. Hadamard factorization for the order-one entire sine function gives a possible exponential multiplier; oddness removes its linear exponent and the derivative at zero fixes its constant. This product turns a Dirichlet oscillator determinant ratio into an elementary sine ratio.