Solution
= Solution
For $n\ge1$, use $\sin^2(nx)=[1-\cos(2nx)]/2$. The <inner product> is
$$
\boxed{\langle f_n,f_n\rangle=\int_{-\pi}^{\pi}\sin^2(nx)\,dx=\pi\quad(n\ge1).}
$$
The stated nonnegative index range also includes $n=0$. Since $f_0$ is the <zero function>, $\langle f_0,f_0\rangle=0$, so it must be removed from any proposed <basis>.