How can I show that the inverse of the induced metric $h_{alpha beta}$ is $h^{alpha beta}$?

So I was reading through Becker, Becker, Schwarz and there is a line in the second chapter that states that $h^{alpha beta} = (h_{alpha beta})^{-1}$ where $h_{alpha beta}$ is defined as:
$$h_{alpha beta} = frac{partial X^{mu}}{partial sigma^{alpha}} frac{partial X^{nu}}{partial sigma^{beta}} g_{mu nu}$$
where $X^{mu}$ is our coordinates on our spacetime manifold, $sigma^{alpha}$ is our coordinates on our worldsheet, and $g_{mu nu}$ is our spacetime metric. This seems very natural given that $h_{alpha beta}$ is precisely the induced metric on the worldsheet and for metrics on our spacetime $g_{mu nu}^{-1} = g^{mu nu}$. However, I am having a hard time proving this. Namely,
$$h_{alpha beta}h^{alpha gamma} = frac{partial X^{mu}}{partial sigma^{alpha}} frac{partial X^{nu}}{partial sigma^{beta}} g_{mu nu}frac{partial X^{mu’}}{partial sigma_{alpha}} frac{partial X^{nu’}}{partial sigma_{gamma}} g_{mu’ nu’}$$
which doesn’t look like it will yield $delta_{beta}^{gamma}$. I have tried fiddling with the algebra with no avail…