An alternate formula for the covariance is \(\mathrm{cov}(X, Y) = \mathbf{E}\left[X(Y - \mu_Y)\right]\);
this property can be used to prove that the covariance is bilinear:
$$\mathrm{cov}(aX + bY, Z) = a \, \mathrm{cov}(X, Z) + b \, \mathrm{cov}(Y, Z). $$