Cardinal collapsing and product forcing

Suppose $\kappa$ is a singular strong limit cardinal of countable cofinality and let $\langle \kappa_{n}: n<\omega \rangle$ be an incrasing sequence of regular cardinals cofinal in $\kappa$. We show that if $cf(2^\kappa)= \kappa^+$, then forcing with the full product $\prod_{n<\omega}Add(\kappa_n,1)$ 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 $$\prod_{n<\omega}Add(\kappa_n,1)/fin \simeq Add(\kappa^+, 1).$$