17. Infinite product formulae

For applications to special functions, the following infinite product formulae for trigonometric functions are useful:^{[citation needed]}

${\displaystyle {\begin{aligned}\sin x&=x\prod _{n=1}^{\infty }\left(1-{\frac {x^{2}}{\pi ^{2}n^{2}}}\right)\\\sinh x&=x\prod _{n=1}^{\infty }\left(1+{\frac {x^{2}}{\pi ^{2}n^{2}}}\right)\\{\frac {\sin x}{x}}&=\prod _{n=1}^{\infty }\cos {\frac {x}{2^{n}}}\end{aligned}}\ \,{\begin{aligned}\cos x&=\prod _{n=1}^{\infty }\left(1-{\frac {x^{2}}{\pi ^{2}(n-{\frac {1}{2}})^{2}}}\right)\\\cosh x&=\prod _{n=1}^{\infty }\left(1+{\frac {x^{2}}{\pi ^{2}(n-{\frac {1}{2}})^{2}}}\right)\\|\sin x|&={\frac {1}{2}}\prod _{n=0}^{\infty }{\sqrt[{2^{n+1}}]{\left|\tan \left(2^{n}x\right)\right|}}\end{aligned}}}$

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