Cosine of Zero is One

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Theorem

$\cos 0 = 1$

where $\cos$ denotes the cosine.


Proof

Recall the definition of the cosine function:

$\ds \cos x = \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{2 n} } {\paren {2 n}!} = 1 - \frac {x^2} {2!} + \frac {x^4} {4!} - \cdots$


Thus:

$\cos 0 = 1 - \dfrac {0^2} {2!} + \dfrac {0^4} {4!} - \cdots = 1$

$\blacksquare$


Also see


Sources

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