Equivalence between covariance definitions

Question

Could you tell me why the following definition of the covariance:

$$ Cov(X, Y) = \mathbb{E} \Big[ \big(X - \mathbb{E}[X]\big) \, \big(Y - \mathbb{E}[Y]\big) \Big] $$

is equivalent to the following one?

$$ Cov(X, Y) = \mathbb{E}[X \, Y] - \mathbb{E}[X] \, \mathbb{E}[Y] $$

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EDIT

By expanding, I get:

\begin{align} Cov(X, Y) &= \mathbb{E}[X \, Y] - \mathbb{E}[X \, \mathbb{E}[Y]] - \mathbb{E}[Y \, \mathbb{E}[X]] + \mathbb{E}[\mathbb{E}[X] \, \mathbb{E}[Y]] = \\ &= \mathbb{E}[X \, Y] - \mathbb{E}[Y] \, \mathbb{E}[X] - \mathbb{E}[X] \, \mathbb{E}[Y] + \mathbb{E}[X] \, \mathbb{E}[Y] = \\ &= \mathbb{E}[X \, Y] - \mathbb{E}[Y] \, \mathbb{E}[X] \end{align}

Answer 1

By expanding:

\begin{align} Cov(X, Y) &= \mathbb{E}[X \, Y] - \mathbb{E}[X \, \mathbb{E}[Y]] - \mathbb{E}[Y \, \mathbb{E}[X]] + \mathbb{E}[\mathbb{E}[X] \, \mathbb{E}[Y]] = \\ &= \mathbb{E}[X \, Y] - \mathbb{E}[Y] \, \mathbb{E}[X] - \mathbb{E}[X] \, \mathbb{E}[Y] + \mathbb{E}[X] \, \mathbb{E}[Y] = \\ &= \mathbb{E}[X \, Y] - \mathbb{E}[Y] \, \mathbb{E}[X] \end{align}