\[\frac {d(Cosx)} {dx} = -Sinx \]
\[Cosx = \sum_{n=0}^{\infty} (-1)^n. \frac{x^{2n}}{(2n)!}\]
\[\frac {d(Cosx)} {dx} = \frac{d}{dx} \sum_{n=0}^{\infty} (-1)^n. \frac{x^{2n}}{(2n)!}\]
\[ = \sum_{n=0}^{\infty} (-1)^n. 2n. \frac{x^{2n -1 }}{(2n)!}\] \[ = -x + \frac{x^3}{3!} - \frac{x^5}{5!} + \frac{x^7}{7!} - ... \] \[ = -[x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + ... ]\]
\[ = -Sinx\]