Well from observation we know $ E[z] = E[w] 0 $ due to them being periodic.
We also know that $ E[x^2]=\sigma_x $ and $ E[Y^2] =\sigma_y $.
So then... $ Var[z] = E[z^2] - (E[z])^2 $ and since $ E[z]=0 $ $ Var[z] = E[z^2] = (x\cos(\theta)+sin(\theta))^2 $
