Euler Formula for Sine Function/Real Numbers/Proof 2

Theorem

 * $\displaystyle \sin x = x \prod_{n \mathop = 1}^\infty \left({1 - \frac {x^2} {n^2 \pi^2}}\right)$

for all $x \in \R$.

Proof
Using De Moivre's Formula:


 * $\sin x = \dfrac {\left({\cos \dfrac x n + i \sin \dfrac x n}\right)^n - \left({\cos \dfrac x n - i \sin \dfrac x n}\right)^n} {2i}$

The difference between two $n$th powers can be extracted into linear factors using $n$th roots of unity.

For large $n$, we can replace:
 * $\cos \dfrac x n$ by $1$


 * $\sin \dfrac x n$ by $\dfrac x n$

He proved it in vol. 1 of his 1748 work Introductio in analysin infinitorum using De Moivre's Formula as above.