Transpose of inverse vs inverse of transpose

Is $(A^{-1})^T = (A^T)^{-1}$ you ask.

Well
$$\begin{align} A^T(A^{-1})^T = (A^{-1}A)^{T} = I^T = I \\ (A^{-1})^TA^T = (AA^{-1})^{T} = I^T = I \end{align}$$

This proves that the inverse of $A^T$ is $(A^{-1})^T$. So the answer to your question is yes.

Here I have used that
$$A^TB^T = (BA)^T.$$
And we have used that the inverse of a matrix $A$ is exactly (by definition) the matrix $B$ such that $AB = BA = I$.