Applying Maclaurin's theorem to the cosine and sine functions, we get

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

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

For both series, the ratio of the nth 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

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

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

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