(A)Let's consider two cases.
W-True.This makes both LHS and RHS True.
W-False.The value of LHS depends upon the truth value of $\forall$xP(x). Same will be the case for RHS.
Hence LHS =RHS.
(B)Using analogy in (A), we can prove that this is also valid.
W-True.LHS=RHS=True
W-False.LHS=RHS=False always.
(C)$\forall x(P(x) \rightarrow W) \equiv \forall xP(x) \rightarrow W$
LHS can be re-written as $\forall x(\lnot P(X) \lor W) \equiv \exists xP(x) \rightarrow W$
(C) is not logically valid.
(D)$\exists x(P(x) \rightarrow W) \equiv \exists x(\lnot P(X) \lor W) $
Using Demorgan law for quantifiers we can again rewrite it as:
$\lnot \forall x P(x) \lor W \equiv \forall x P(x) \rightarrow W$
Option (D) is valid.