We work over $k=\mathbb{C}$. We consider the 
the Grassmanian $G(2,4)$ of lines in $\mathbb P^3$ which we embed
by Plücker into $\mathbb P^5$. It is basic that under this embedding 
$G(2,4)$ is isomorphic with a $4$-dimnl quadric $Q^4 \subset \mathbb P^5$.


Now, take any curve $C'$ in $\mathbb P^3$ and associate to it a 
curve $C$ in $G(2,4)$  consisting of the collection of lines tangent to $C'$. 
In [this answer][1] to https://mathoverflow.net/questions/69885/quadrics-containing-the-tangential-variety-of-a-curve Dmitri Panov claims that after this Plücker identification of $G(2,4)$ with $Q^4$ it can be shown that the tangent variety $TC$ of $C$ is contained
in $Q^4$. 

Question: How to see it? Is it possible to prove it purely algebraically?
Indeed as sketched in the comments below the linked answer it's possible
to show it analytically working with local parametrization of the curve $C'$.
But I'm wondering isn't it possible to give a pure algebraic proof of it?
At all, that seems not to be hard, but I can't fiddle it out.


  [1]: https://mathoverflow.net/q/69894