I think the problem lies mainly in your second line which is correct but confusing and hard to work with.
Much better I think to just note that
sin[2 theta] = -1/2 at
(2 theta) = (11 pi)/6 + 2 pi k, and (7 pi)/6 + 2 pi k, k an integer
Now you have two simple equations to solve w/o all the fuss about even vs. odd n etc.
(sorry LaTex doesn't seem to be working today)
comparing solutions
theta = (11 pi)/12 + pi k = -pi/12 + pi k, i.e your first solution
theta = (7 pi)/12 + pi k, i.e. your second solution
so we ended up with the same answers but I believe my solution is a bit more straightforward than yours.
You did get the correct answers though.