Derivative of Sine Function/Proof 1

Proof
From the definition of the sine function, we have:
 * $\displaystyle \sin x = \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{2 n + 1} } {\paren {2 n + 1}!}$

From Radius of Convergence of Power Series over Factorial, this series converges for all $x$.

From Power Series is Differentiable on Interval of Convergence:

The result follows from the definition of the cosine function.