Covariance depending on wether the random variables are equally distributed

Question

X and Y are two random variables that have the same distribution and they are not independent. True or false,

Cov(X+Y,X-Y)=0

So I used the properties of Covariance and ended up with Var(X)-Var(Y)...Im not sure if this is equal to 0. If they tell you that the random variables are equally distributed, does this mean that they have the same parameters or that they simply follow a certain type of distribution (say Normal, Poisson, etc). Thanks in advance.

Answer 1

"$X$ and $Y$ have the same distribution" means literally that. For example, if $X$ is Poisson with mean $4$, then $Y$ is also Poisson with mean $4$. If $X$ is uniform over the values $\{1,4, 7\}$, then $Y$ is also uniform over those values. And so on.

In particular, $X$ and $Y$ have the same expectation, the same variance,...