I have the following problem of the book "Massey, Algebraic Topology an Introduction" (Chapter 4, section 4, problem 1):
Let $X$ arcwise connected and $U,V$ open arcwise connected subsets of X. Suppose that $U\cup V=X$, $U\cap V$ non-empty and arcwise connected.
Consider $\varphi_1:\pi(U\cap V)\to\pi(U)$, $\varphi_2:\pi(U\cap V)\to\pi(V)$, $\psi_1:\pi(U)\to\pi(X)$, $\psi_2:\pi(V)\to\pi(X)$, all morphism induced by the inclusions.
Prove that if $\varphi_2$ is isomorphism onto, then so is $\psi_1$.
I tried to use the first version of Seifert-Van Kampen Theorem but I have no result. Maybe is a wrong way to do this problem.
Any solution or hint would be appreciated.
Thanks in advance.