Trigonometry/Power Series for Cosine and Sine

Applying Maclaurin's theorem to the cosine and sine functions for angle x (in radians), we get

$\cos (x)=1-\frac{x^2}{2!}+\frac{x^4}{4!}-\cdots =\sum _{n=0}^{\mathrm{\infty }}\frac{(-1{)}^n{x}^{2n}}{(2n)!}$

$\sin (x)=x-\frac{x^3}{3!}+\frac{x^5}{5!}-\cdots =\sum _{n=0}^{\mathrm{\infty }}\frac{(-1{)}^n{x}^{2n+1}}{(2n+1)!}$

For both series, the ratio of the $n^{th}$ to the $(n-1{)}^{th}$ term tends to zero for all $x$. Thus, both series are absolutely convergent for all $x$.

Many properties of the cosine and sine functions can easily be derived from these expansions, such as

$\sin (-x)=-\sin (x)$

$\cos (-x)=\cos (x)$

$\frac{d}{dx}\sin (x)=\cos (x)$