The formula for \(\operatorname{cov}\) is:
\[ \operatorname{cov}(X, Y) = E[(X - E[X])(Y - E[Y])] \]
Expanding:
\[ \operatorname{cov}(X, Y) = E[(XY - XE[Y] -YE[X] + E[X]E[Y])] \]
Using linearity of expectation property:
\begin{align} \operatorname{cov}(X, Y) &= E[XY] - E[XE[Y]] - E[YE[X]] + E[E[X]E[Y]] \\ &= E[XY] - E[Y]E[X] - E[X]E[Y] + E[X]E[Y] \\ &= E[XY] - E[X]E[Y] \end{align}