confusion about equivalent covariance formulas

Question

$Cov(X,Y)$ = $E[$((X-E(X))((Y-E(Y))]$ = $E(XY)-E(X)E(Y)$ = $E[XY-E(X)Y]$ = $E[(X-E(X))Y]
So, $E[((X-E(X))(Y-E(Y))]$ = $E[((X-E(X))Y]$ . But intuitively, $Y$ and $Y-E(Y)$ are simply different. How should I understand this equivalence? Thank you.

Answer 1

It boils down to this: if $Z$ has mean $0$, then for any r.v. $S$ and any constant $a$, we have $$ E[Z(S-a)]=E[ZS-Za]=E[ZS]-E[Za]=E[ZS]-a\underbrace{E[Z]}_0=E[ZS]. $$ In this case, $Z=X-E(X)$, $S=Y$, and $a=E(Y)$.