Fourier Series of $f$ on the given interval

Fourier Series of $f$ on the given interval

my goal is to find the Fourier series of f on the given interval:
$$f(x) = \begin{cases} 0, & \text{if } -\pi < x < 0 \\ \sin(x), & \text{if
} 0 \le x < \pi \end{cases}$$
I know there aren't any $\sin(nx)$ in the series but i'm having trouble
with the $\cos(nx)$. this is my integral:
$$\frac1\pi\int_0^\pi
\sin(x)\cos(nx)=\frac{\cos(nx)+1}{\pi(1-n^2)}=\frac{(-1)^n+1}{\pi(1-n^2)}$$
normaly i simply take the result and put it in the formula yet i don't
know how to bring the result to this form:
$$\sum_{i=2}^\infty (\text{something}) \cos(nx)$$
please not it starts from 2 (i never encountered a task which it starts
from 2).... any ideas what shall i do next?