Find a Maclaurin series for sin(2+x)? Trouble finding a representation for it?

Question

Here's the work i've done to find the Maclaurin series. However, I'm having a very hard time finding a representation for the series using sum, n for the nth term, and x from g(x).

Answer 1

One way to get the "${+}{+}{-}{-}$" pattern is with $(-1)^{n(n-1)/2}$.

To alternate between $\sin(2)$ and $\cos(2)$: $$\sin(2)\frac{(-1)^n+1}{2}+\cos(2)\frac{(-1)^{n-1}+1}{2}$$

So you could write $$\sum_{n=0}^{\infty}(-1)^{n(n-1)/2}\left(\sin(2)\frac{(-1)^n+1}{2}+\cos(2)\frac{(-1)^{n-1}+1}{2}\right)\frac{x^n}{n!}$$