[Math] Prove that d/dx (sin x) = cos x, using Taylor series

Question

Show by differentiation of the series for sin x that

$$\frac{d}{dx} (\sin x) = \cos x$$ (Using Taylor series.)

If you can given an indication or solved answer to my question would be great.

Thanks

Answer 1

Note that $$\dfrac{d}{dx}\left(\dfrac{x^{2k+1}}{(2k+1)!}\right) = \dfrac{x^{2k}}{(2k)!}$$