Let X be an algebraic variety over an algebraically closed field. Let Y be an irreducible subvariety of X of codimension 1 with generic point y. The structure of the tangent cone \(T_{X,x}\) of X at almost all closed points x in Y is compared to that of \(T_{x,y}\). This is done by using the theory of branches as developed by \textit{S. Greco} in Proc. internat. Symp. Algebraic Geometry, Kyoto 1977, 477-493 (1977; Zbl 0411.14001) and showing a bijection between the branches at x (resp. components of \(T_{X,x})\) and the geometric branches at y (resp. geometric components of \(T_{X,y})\).