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In Bott and Tu's book, "Differential forms in Algebraic Topology", page 45, Section 5 of Chapter one, he tried to prove the Poincare duality. But I find one step mysterious, namely when he describes the sequence $H_c^{n-q-1}(U\cup V) \rightarrow^{d_*} H_c^{n-q}(U\cap V) \rightarrow H_c^{n-q}(U) \oplus H_c^{n-q}(V)$, and if $\tau$ is in $H_c^{n-q-1}(U\cup V)$, then $d_* \tau$ can be such that

(-(extension by 0 of $d_* \tau$ to $U$), (extension by 0 of $d_* \tau$ to $V$)) = ($d(\rho_U\tau), d(\rho_V\tau))$

where $\rho_U$ and $\rho_V$ are partition of unity. I feel confused by this part, why this is true?

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The result is given by diagram chasing.

The map $d_*$ is the connecting homomorphism, so by chasing it we shall do the following:

1.Find inverse image in $\Omega^*_c(U) \oplus \Omega^*_c(V)$.

2.differentiate it.

3.Find the inverse image in $\Omega^*_c(U \cap V)$

And you can check proof of Prop 2.7 on page 26 for the explicit maps in each step.

I should make it a comment but I do not have enough reputation. Sorry about that.

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