Define the covariance, $\operatorname{cov}(X, Y)$, of two random variables $X$ and $Y$.

Prove, or give a counterexample to, each of the following statements.

(a) For any random variables $X, Y, Z$

$$\operatorname{cov}(X+Y, Z)=\operatorname{cov}(X, Z)+\operatorname{cov}(Y, Z)$$

(b) If $X$ and $Y$ are identically distributed, not necessarily independent, random variables then

$$\operatorname{cov}(X+Y, X-Y)=0$$