Taylor Series of trigonometric function I searched quite a bit online but could only find the MACLAURIN SERIES of $\sin x$ and $\cos x$: 
$$\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!}+\cdots$$ 
$$\cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!}+\cdots$$
Can anyone explain how we can express the TAYLOR SERIES of $\sin x$ and $\cos x$, and also show me the derivation? 
Thanks in advance! 

Observe
\begin{align}
\sin x =&\ \sin(x-a+a) = \sin(x-a) \cos a +\cos(x-a) \sin a\\
=&\ \cos a \sum^\infty_{n=0} (-1)^n\frac{(x-a)^{2n+1}}{(2n+1)!} + \sin a\sum^\infty_{n=0}(-1)^n\frac{(x-a)^{2n}}{(2n)!}.
\end{align}
I will leave it as an exercise for the reader to figure out the Taylor expansion for $\cos x$ at $x=a$. 
