Consider a curve $X$ (over a field $k$), which is an union of two smooth and connected curves $C$ and $D$ meeting at exactly one point $P \in X$ transversally. Let $E$ be a vector bundle on $X$ such that $E_{| C}$ is a trivial vector bundle.
Q: Is there an open $U \subset X$ containing $C$ such that $E_{|U}$ is trivial?
Ansatz: Take a neighborhood $V \subset X$ of $P$ such that $E$ trivializes over $V$. Then $U = V \cup C$ is open in $X$. On the other hand, $C \setminus \{P\}$ is open in $X$. These two sets has the intersection $V \cap C \subset C$. Therefore, since $E_{|C}$ and $E$ have the same transition functions on this intersection, it follows that $E$ is trivial on $U$ using the assumption that $E_{|C}$ is trivial, whence it has transition functions equivalent to the trivial transition functions.