Smooth submanifold of $C^n$

Question

How can I prove that a subset of $C^n$ given by the following equation:
$ Z^2_1+Z^2_2+ .... +Z^2_n = 1$ is a smooth submanifold of $C^n$, which is diffeomorphic to the tangent space of an $(n-1)$ dimensional sphere, $S^{n-1}$? Am I to prove that the sphere is diffeomorphic to it's own tangent space? Because I couldn't quite understand the problem itself.