For applications to special functions, the following infinite product formulae for trigonometric functions are useful:[citation needed]
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Multiple-angle formulae Double-angle, triple-angle, and half-angle formulae Double-angle formulae Triple-angle formulae ...
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These are also known as the addition and subtraction theorems or formulae . The identities can be derived by combining right triangles ... -
By examining the unit circle, the following properties of the trigonometric functions can be established. Symmetry When the trigon...
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c8c14aaad3e338d44553e8762837af4d44498529)
