Prove That $1/\sin(i \pi/n)+1/\sin(j \pi/n) \ne 2$ When $n$ is Odd

Prove That $1/\sin(i \pi/n)+1/\sin(j \pi/n) \ne 2$ When $n$ is Odd

The original problem is: Prove that $1/\sin(i \pi/n)+1/\sin(j \pi/n) \ne
2$ when $n$ is odd.
I tried, and found that although everything looks similar, I actually know
nothing about this kind of problem. Even whether $\sin(i \pi/n)+\sin(j
\pi/n)$ would equal to $1/2$ or something.
It looks like algebraic number theory, but every rational number is in all
number fields, so I don't know how to make the problem fit into this
framework.
So my problem is:
How to Solve the Original Problem?
What values can $\sum_n a_i\sin(i \pi/n)$ express ($a_i,i,n\in
\mathbb{Z}$) (This one is somewhat vague)?
Is there any general idea behind these problems?