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\cos X & = I - \frac{X^2}{2!} + \frac{X^4}{4!} - \frac{X^6}{6!} + \cdots & = \sum_{n=0}^\infty \frac{(-1)^n}{(2n)!}X^{2n}
\end{align}</math>
with {{math|''X<sup>n</sup>''}} being the {{mvar|n}}th [[Matrix multiplication#Powers of
Equivalently, they can be defined using the [[matrix exponential]] along with the matrix equivalent of [[Euler's formula]], {{math|''e<sup>iX</sup>'' {{=}} cos ''X'' + ''i'' sin ''X''}}, yielding
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