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\begin{align}
&\bbox[10px,#ffd]{\ds{%
\prod_{n = 1}^{\infty}\bracks{1 - {2 \over \pars{2n + 1}^{2}}}}} =
\prod_{n = 1}^{\infty}\bracks{1 - {1/2 \over \pars{n + 1/2}^{2}}}
\\[5mm] = &\
\lim_{N \to \infty}
{\bracks{\prod_{n = 1}^{N}\pars{n + 1/2 - \root{2}/2}}
\bracks{\prod_{n = 1}^{N}\pars{n + 1/2 + \root{2}/2}} \over
\bracks{\prod_{n = 1}^{N}\pars{n + 1/2}}^{2}}
\\[5mm] = &\
\lim_{N \to \infty}
{\pars{3/2 - \root{2}/2}^{\overline{N}}\pars{3/2 + \root{2}/2}^{\overline{N}} \over \bracks{\pars{3/2}^{\overline{N}}}^{2}}
\\[5mm] = &\
{\Gamma^{2}\pars{3/2} \over
\Gamma\pars{3/2 + \root{2}/2}\Gamma\pars{3/2 - \root{2}/2}}\,
\lim_{N \to \infty}
{\pars{N + 1/2 - \root{2}/2}!\pars{N + 1/2 + \root{2}/2}! \over
\bracks{\pars{N + 1/2}!}^{2}}
\\[5mm] = &\
{\pi/4 \over
\pars{-1/4}\Gamma\pars{1/2 + \root{2}/2}\Gamma\pars{1/2 - \root{2}/2}}\
\times
\\[2mm] &\
\lim_{N \to \infty}
{\bracks{\pars{N + 1/2 - \root{2}/2}^{N + 1 - \root{2}/2}}
\bracks{\pars{N + 1/2 + \root{2}/2}^{N + 1 + \root{2}/2}}
\expo{-2N - 1} \over
\pars{N + 1/2}^{2N + 2}\expo{-2N - 1}}
\\[5mm] = &\
-\sin\pars{\pi\bracks{{1 \over 2} + {\root{2} \over 2}}}
\times
\\[2mm] &\
\lim_{N \to \infty}
{\bracks{1 + \pars{1/2 - \root{2}/2}/N}^{N}
\bracks{1 + \pars{1/2 + \root{2}/2}/N}^{N}
\over
\braces{\bracks{1 + \pars{1/2}/N}^{N}}^{2}}
\\[5mm] = &
-\cos\pars{\pi\,{\root{2} \over 2}}\,
{\exp\pars{1/2 - \root{2}/2}\exp\pars{1/2 \root{2}/2} \over \bracks{\exp\pars{1/2}}^{2}} = \bbx{-\,\cos\pars{{\root{2} \over 2}\,\pi}}
\approx 0.6057
\end{align}