$$ \begin{eqnarray} Cov(X,Y) &=& E[(X-\mu_X)(Y-\mu_Y)] \\ &=& E[(XY - \mu_XY - X\mu_Y + \mu_X\mu_Y)]\\ &=& E[XY] - E[\mu_XY] - E[X\mu_Y] + E[\mu_X\mu_Y]\\ &=& E[XY] - \mu_XE[Y] - E[X]\mu_Y + \mu_X\mu_Y\\ &=& E[XY] - E[X]E[Y] - E[X]E[Y] + E[X]E[Y]\\ &=& E[XY] - E[X]E[Y] \end{eqnarray} $$