Please see attached picture. I completed question 2 successfully (I think), but am having trouble with question 3.

It seems question 2 came out nicely because there were some trig identities I could use to cancel out terms. However, this does not appear to be the case with 3. I got that the basis is not orthonormal, however, it seems like the components of the metric tensor will be huge long expressions. Is this the case, or am I missing a trig identity here?

Any help would be appreciated and thanks in advance!