Suppose $X\subset \Bbb P^n$ is a projective variety over an algebraically closed field (in the sense of Hartshorne Chapter I) and $P$ is a fixed point in $\Bbb P^n\setminus X$. Why should it be the case that for generic line $L$ through $P$ meeting $X$, we have that the line $L$ is not in the tangent space $T_QX$ for any point $Q\in L\cap X$? It's "geometrically obvious" to me that it must be true, but I cannot quite put together a full proof. Here are some of the things I've tried:
Restricting to the copy of $\Bbb A^1$ given by $L\setminus\{P\}$, this condition should be expressible as "the principal ideal cutting out our closed subset is generated by a polynomial with no multiple roots", but I'm not sure how to make this something I can actually analyze.
Using an incidence correspondence. There should be some way to embed $X^{sm}$ in to something times the Grassmanian parameterizing the tangent planes (?), take the closure, and then things pop out. This idea is inspired by this post, and I am really flailing at this one.
I would like to avoid using big theorems not present in some form in Hartshorne for this, though if you have a particularly nice solution which involves such a thing, please do go ahead and post it anyways. (In particular, if you want to use the characterization of degree as "number of intersection points with a generic linear subspace of complimentary dimension", I would like to see a proof or reference to a proof that this is the same as Hartshorne's characterization of degree via Hilbert polynomials.)