How do you prove the following statement?

"If $S \subseteq cof(<\kappa) \cap \kappa^+$ is stationary, and $\kappa^{<\kappa} =\kappa$, then the stationarity of $S$ Is preserved by $\kappa$-closed forcing."

If we just assume $\kappa$ is regular, do we need the cardinal arithmetic?

The following statement can be proven using elementary submodels and sufficiently generic conditions:

"If $S \subseteq cof(<\kappa) \cap \kappa^+$ is stationary, and $\kappa^{<\kappa} =\kappa$, then the stationarity of $S$ is preserved by $\kappa$-closed forcing."

If we just assume $\kappa$ is regular, do we need the cardinal arithmetic?