Euler Formula for Cosine Function

Proof
We have that $\cos x$ has a power series representation:


 * $\cos x = 1 - \dfrac {x^2} {2!} + \dfrac {x^4} {4!} - \dfrac {x^6} {6!} + \dotsb$

The roots of cosine are the numbers $\paren {k + \dfrac 1 2} \pi$, where $k$ is any integer.

From the Polynomial Factor Theorem, the following (after simplification) might be true:


 * $\ds \cos x = A \prod \paren {1 - \frac {2 x} {\paren {2 k + 1} \pi} }$

where the product is taken over all $k \in \Z$, and $A$ is some constant.

The intuition is as follows.

Letting $x$ tend to $0$ in the above equation implies that $A = 1$.

It remains to formalize the above claims.

Also see

 * Euler Formula for Sine Function