Given a manifold $M$ and opens sets $U_1,U_2 \subseteq M$ such that $M=U_1 \cup U_2$, the Connecting Homomorphism for Mayer-Vietoris sequence is defined as $$\delta: H^p(U_1 \cap U_2) \rightarrow H^{p+1}(M) \\ [\omega] \mapsto \begin{cases} [-d(\rho_2 \omega)] \quad\text{on}\ U_1 \\ [d(\rho_1 \omega)] \quad \text{on} \ U_2 \end{cases}$$ where $\{\rho_j\}_{j=1,2}$ is a partition of unity subordinate to $\{U_j\}_{j=1,2}$.
So $d(\rho_j \omega)$ is an exact $(p+1)$-form for $j=1,2$. Why its cohomology class is not zero on $H^{p+1}(M)$?