You have to do it for arbitrarily different eigenvalues. So, do the second, which will give you that the eigenfunctions corresponding toDo they want me to choose two values such as n=1, and n=2, compute their respective eigenfunctions and see if they are orthogonal?

Or do I have to do it for case such as, n=a and n=b?anytwo distinct eigenvalues are orthogonal.

Technically, the inner product will beAlso, how do I compute the integral to verify that it's 0? Isn't it just going to be sin(something)*sin(somethingdifferent)?

Assume and show that the integral is zero. Then you're done. Make sense?