This is probably a very easy question, but I can't find the answer to it..
I'm working on Fourier coefficients and whether or not the integrals become zero. As far as i'm concerned this integral below equals zero due to it being odd over the interval -pi to pi. That is fair, but is it an easy way of noticing that the function is odd when it has 3 terms or more?
If $f(x)$ is odd and $g(x)$ is even (or visa versa) then $f(x)g(x)$ is odd.
If $f(x)$ is even and $g(x)$ is even, then $f(x)g(x)$ is even.
If $f(x)$ is odd and $g(x)$ is odd, then $f(x)g(x)$ is even.
This particular example can be seen by using the formula: $$\sin(a)\cos(b) = \frac{1}{2}\left(\sin(a+b)-\sin(a-b)\right)$$
Then $$\sin(10x)\cos(x)=\frac{1}{2}\sin(11x)-\frac{1}{2}\sin(9x)$$
Which gives us the Fourier series for $\sin(10x)\cos(x)$, which contains no $\cos$ terms.