Derive Maclaurin series for $\sin x$ and $\cos x$ (#452)

Derive Maclaurin series for $\sin x$ and $\cos x$ (#452)

sin⁡x=∑n=0∞(−1)nx2n+1(2n+1)!\sin x = \sum_{n=0}^\infty \frac{(-1)^n x^{2n+1}}{(2n+1)!}, cos⁡x=∑n=0∞(−1)nx2n(2n)!\cos x = \sum_{n=0}^\infty \frac{(-1)^n x^{2n}}{(2n)!}.