\(\frac{d(cosx)}{dx} = -{sinx}\)
\(cosx = \sum\limits_{n=0}^{\infty} {(-1)^n} . \frac{(x)^{2n}}{(2n)!}\)
\(\frac{d(cosx)}{dx} = \frac{d}{dx} \sum\limits_{n=0}^{\infty} {(-1)^n} . \frac{(x)^{2n}}{(2n)!}\)
\(= \frac{d}{dx} \sum\limits_{n=0}^{\infty} {(-1)^n} .2n \frac{(x)^{2n-1}}{(2n)!}\)
\(= -x + \frac{x^3}{3!} - \frac{x^5}{5!} + \frac{x^7}{7!}- ...\)
\(= -[x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7! + ...}\)
\(= -sinx\)