Chapter 12.7, Problem 1P

By a method similar to that we used to show that the $P_l$ ’s are an orthogonal set of functions on $(-1,1),$ show that the solutions of $y_n^{\prime \prime }=-n^2{y}_n$ are an orthogonal set on $(-\pi ,\pi ).$ Hint: You should know what functions the solutions $y_n$ are; do not use the functions themselves, but you may use their values and

the values of their derivatives at $-\pi$ and $\pi$ to evaluate the integrated part of your equation.