I have some questions.

The first one is about the product of Prikry's forcing. Let $\kappa$ be a measurable cardinal, $U_1, U_2$ be normal measures on $\kappa$ and let $\mathbb{P}_{U_1}, \mathbb{P}_{U_2}$ be the corresponding Prikry forcings. Let $G\times H$ be $\mathbb{P}_{U_1}\times \mathbb{P}_{U_2}-$generic over $V$. It is easily seen that in the extension there are new subsets of $\omega$ (for example if $(x_n: n<\omega), (y_n: n<\omega)$ are the Prikry sequences added by $G, H$ respectively, then $\{ n<\omega: x_n < y_n\}$ is such a set).

Question 1.1Is $\kappa$ preserved in the extension $V[G\times H]$? Do $V$ and $V[G\times H]$ have the same cardinals?

**Remark.** Though the question remained unanswered in general, but by Yair Hayut's very nice answer, given a normal measure $U$ on $\kappa, \mathbb{P}_U^2$ does not collapse cardinals. His proof extends easily to show that for any natural number $n>1, \mathbb{P}_U^n$ is forcing isomorphism to $\mathbb{P}_U\times \mathbb{C},$ hence it preserves cardinals.

Question 1.2.What is the least cardinal $\lambda$ such that $\mathbb{P}_U^\lambda$ collapses some cardinals? What is the least cardinal $\delta$ such that $\mathbb{P}_U^\delta$ collapses $\kappa$? Are $\lambda$ and $\delta$ equal?

My second question is about Cohen forcing. Let $\kappa$ be a Mahlo cardinal, let $\mathbb{P}$ be the reverse Easton iteration of $Add(\alpha,1)$ for all inaccessible cardinals $\alpha\leq \kappa,$ and let $G$ be $\mathbb{P}-$generic over $V$.

Question 2.Suppose $\alpha$ is an inaccessible cardinal $\leq \kappa.$ Is there an $H\in V[G]$ which is $Add(\alpha,1)^V-$generic over $V$? (of course the answer is yest for the least inaccessible).