I am looking for a proof/reference of the following simple fact, which I think it holds true.

Let $S\subset \mathbb{P}^n$ be a surface embedded by a very ample linear system. Then I know that the dual variety $S^*$ is an irreducible hypersurface. Is it true that the generic plane section of $S^*$ is an irreducible curve, smooth except for at most ordinary double points and cusps?

I guess one can translate this in something concerning the generic projection of $S$ onto a $\mathbb{P}^2$.