Chapter 7.11, Problem 10P

A general form of Parseval’s theorem says that if two functions are expanded in Fourier series $\begin{array}{l}f(x)=\frac{1}{2}{a}_0+{\displaystyle \underset{1}{\overset{\infty }{\sum }}{a}_n}\cos nx+{\displaystyle \underset{1}{\overset{\infty }{\sum }}{b}_n}\sin nx\hfill \\ g(x)=\frac{1}{2}{a}_0^{\prime }+{\displaystyle \underset{1}{\overset{\infty }{\sum }}{a}_n^{\prime }}\cos nx+{\displaystyle \underset{1}{\overset{\infty }{\sum }}{b}_n^{\prime }}\sin nx\hfill \end{array}$ then the average value of $f(x)g(x)$ is $\frac{1}{4}{a}_0{a}_0^{\prime }+\frac{1}{2}{\displaystyle \underset{1}{\overset{\infty }{\sum }}{a}_n}{a}_n^{\prime }+\frac{1}{2}{\displaystyle \underset{1}{\overset{\infty }{\sum }}{b}_n}{b}_n^{\prime }.$ Prove this.