I am doing exercise 21 of section 1.2 in Hatcher's Algebraic Topology.
Show that the join $X\ast Y$ of two nonempty spaces $X$ and $Y$ is simply connected if $X$ is path connected.
I proved that $X\ast Y$ is path connected via the existence of paths between $(x,y,t)$ and $(x,y,0)$, and then used the path connectedness of $X$. But, I do not know how to prove that fundamental group is trivial using Van Kampen's theorem.
I tried to cover $X\ast Y$ by $X \times Y \times [0,1)/ \!\sim$ and $X \times Y \times (0,1]/ \!\sim$, but their intersection is not necessarily path connected since $Y$ is not path connected. And I did see this question that proves this statement, but I would like to use Van Kampen's Theorem in the solution instead (after all, this exercise is in the section on Van Kampen's Theorem in Hatcher's book).