Cardinal collapsing and product forcing

Suppose $\kappa$ is a singular strong limit cardinal of countable cofinality and let $⟨{\kappa }_n:n<\omega ⟩$ be an incrasing sequence of regular cardinals cofinal in $\kappa$. We show that if $cf(2^\kappa)= \kappa^+$,thenforcingwiththefullproduct$\prod_{n<\omega}Add(\kappa_n,1)$collapses$2^\kappa$into$\kappa^+$. This result gives a consistent positive answer to a question of Sy Friedman. We also give a new proof of a result due to Shelah by showing that if the sequence carries a scale of length ${\kappa }^{+},$ then forcing with ${\prod }_{n<\omega }Add({\kappa }_n,1)$ adds a generic filter for $Add({\kappa }^{+},1)$, and indeed $$\prod _{n<\omega }Add({\kappa }_n,1)\mathrm{/}fin\simeq Add({\kappa }^{+},1)\mathrm{.}$$