Past exam of the mathematical methods I course of the University of Oxford / 2014 resit / 10 / a

For $n\ge 1$, use ${\sin }^2(nx)=[1-\cos (2nx)]/2$. The inner product isThe stated nonnegative index range also includes $n=0$. Since $f_0$ is the zero function, $⟨f_0,f_0⟩=0$, so it must be removed from any proposed basis.

The product-to-sum formula givesWhen both indices are positive the two integer frequencies integrate to zero. If either index is zero the integrand already vanishes. This is Fourier orthogonality.