I tried to do this by contradiction.
So we have that $(X, \tau_2)$ is Hausdorff, and $\tau_2 \subset \tau_1$.
Suppose that $(X, \tau_1)$ was not Hausdorff. Then we have elements $y,z \in X$ where $z\neq y $ and two open sets, call them $U$ and $W$ where $y \in U$, $z \in W$ and $U \cap W \neq \emptyset$.
I'm kinda stuck here, I wanna use that $\tau_1$ is finer than $\tau_2$ somehow, but I can't really figure it out.
Any help would be great!