It took a little while but I realized the sigma expressions are different from line to line.

Try using the fact that sigma(x - x_bar)^2 = sigma(xi^2) - (sigma(xi))^2 and that sigma(xi) = N*x_bar.

To start you off, you have:

sigma^2[1/N + x_bar^2/[sigma(xi^2) - x_bar^2]]

Now use the hint that [x_bar^2 - sigma(xi^2) + sigma(xi^2)]/[sigma(xi^2) - x_bar^2] = -1 + sigma(xi^2)/[sigma(xi^2) - x_bar^2].