= Solution
Choose the positive local branch of the constraint,
$$
\sigma=\sqrt{1-\boldsymbol\pi^2}.
$$
The <chain rule> gives
$$
\partial_i\sigma=-\frac{\pi_a\partial_i\pi_a}{\sqrt{1-\boldsymbol\pi^2}},
$$
and therefore
$$
(\partial_i\mathbf n)^2=(\partial_i\pi_a)(\partial_i\pi_a)
+\frac{\pi_a(\partial_i\pi_a)\pi_b(\partial_i\pi_b)}{1-\boldsymbol\pi^2}.
$$
Substitution into the original free energy gives exactly
$$
\boxed{F[\boldsymbol\pi]=\int d^dx\,\frac1{2g}\left[(\partial_i\pi_a)(\partial_i\pi_a)+\frac{\pi_a(\partial_i\pi_a)\pi_b(\partial_i\pi_b)}{1-\boldsymbol\pi^2}\right]}.
$$
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