$\newcommand{\bbx}[1]{\,\bbox[15px,border:1px groove navy]{\displaystyle{#1}}\,}
\newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack}
\newcommand{\dd}{\mathrm{d}}
\newcommand{\ds}[1]{\displaystyle{#1}}
\newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,}
\newcommand{\ic}{\mathrm{i}}
\newcommand{\mc}[1]{\mathcal{#1}}
\newcommand{\mrm}[1]{\mathrm{#1}}
\newcommand{\on}[1]{\operatorname{#1}}
\newcommand{\pars}[1]{\left(\,{#1}\,\right)}
\newcommand{\partiald}[3][]{\frac{\partial^{#1} #2}{\partial #3^{#1}}}
\newcommand{\root}[2][]{\,\sqrt[#1]{\,{#2}\,}\,}
\newcommand{\totald}[3][]{\frac{\mathrm{d}^{#1} #2}{\mathrm{d} #3^{#1}}}
\newcommand{\verts}[1]{\left\vert\,{#1}\,\right\vert}$
\begin{align}
&\bbox[5px,#ffd]{\left.\prod_{n = 0}^{m}
\bracks{1 - 4z^{2}/\pars{2n + 1}^{2}}
\right\vert_{\,m\ \in\ \mathbb{N}_{\,\geq 1}}} =
\prod_{n = 0}^{m}
{\pars{n + 1/2}^{2} - z^{2} \over \pars{n + 1/2}^{2}}
\\[5mm] = &\
\on{f}_{m}\pars{z}\on{f}_{m}\pars{-z}
\end{align}
where
\begin{align}
\on{f}_{m}\pars{z} & \equiv \prod_{n = 0}^{m}
{n + 1/2 - z \over n + 1/2} =
{\pars{1/2 - z}^{\overline{m + 1}} \over
\pars{1/2}^{\overline{m + 1}}}
\\[5mm] & =
{\pars{1/2 - z + m}!\,/\,\Gamma\pars{1/2 - z} \over
\pars{1/2 + m}!\,/\,\Gamma\pars{1/2}}
\\[5mm] & \stackrel{{\rm as}\ m\ \to\ \infty}{\sim}\,\,\,
{\root{\pi} \over \Gamma\pars{1/2 - z}}\ \times
\\[2mm] &\
{\root{2\pi}\pars{1/2 - z + m}^{1 - z + m}\,\,\,
\expo{-1/2 + z - m} \over
\root{2\pi}\pars{1/2 + m}^{1 + m}\,\,
\expo{-1/2 - m}}
\\[5mm] & \stackrel{{\rm as}\ m\ \to\ \infty}{\sim}\,\,\,
{\root{\pi} \over \Gamma\pars{1/2 - z}}\ \times
\\[2mm] &\
{m^{1 - z + m}\,\,\,\,\bracks{1 + \pars{1/2 - z}/m}^{\,m}
\over
m^{m + 1}\,\,\,\bracks{1 + \pars{1/2}/m}^{\,m}}\expo{z}
\\[5mm] & \stackrel{{\rm as}\ m\ \to\ \infty}{\sim}\,\,\,
{\root{\pi} \over \Gamma\pars{1/2 - z}}\,m^{-z}
\end{align}
Then,
\begin{align}
&\bbox[5px,#ffd]{\left.\prod_{n = 0}^{m}
\bracks{1 - 4z^{2}/\pars{2n + 1}^{2}}
\right\vert_{\,m\ \in\ \mathbb{N}_{\,\geq 1}}}
\\[5mm] = &\
\lim_{m \to \infty}\braces{\bracks{{\root{\pi} \over \Gamma\pars{1/2 - z}}\,m^{-z}}\bracks{{\root{\pi} \over \Gamma\pars{1/2 + z}}\,m^{z}}}
\\[5mm] = &\
{\pi \over \pi/\sin\pars{\pi\bracks{1/2 + z}}} =
\bbx{\cos\pars{\pi z}} \\ &
\end{align}