$X = Y + Z$ Where $Y$ and $Z$ are independent Gaussians with zero means and known variances.
How do you simplify the left side of this to equal the right side?
I think I see why $Cov(Y+Z,Y+Z)$ equals $\sigma_{Y}^2+\sigma_{Z}^2$, but don't know why its in the denominator or how the numerator is obtained.
It is a mistake somewhere in the book or wherever you got the problem. It should be:
$$\frac{\text{Cov}(Y,Y+Z)}{\text{Cov}(Y+Z,Y+Z)}x=\ldots$$ since: $$\require{cancel}\text{Cov}(Y,Y+Z)=\text{Cov}(Y,Y)+\cancel{\text{Cov}(Y,Z)}=\sigma_Y^2$$ and the denominator is as you said: $$\text{Cov}(Y+Z,Y+Z)=\text{Cov}(Y,Y)+\cancel{\text{Cov}(Y,Z)}+\cancel{\text{Cov}(Z,Y)}+\text{Cov}(Z,Z)=\sigma_Y^2+\sigma_Z^2$$