I will state the question first then my work.
Q:
We have a regression model, $Y_i=\beta_0+\beta_1X_i+\mu_i$ where $Cov(\mu_i,X_i)=0$ is not guaranteed. Suppose that $Z_i$ is an instrumental variable and we have $\hat{\beta_1}$ be the two least squares estimator and we have $\hat{\beta_0}=\bar{Y_n}-\beta_1\bar{X_n}$. Show that $\widehat{Cov(\hat{\mu},Z_i)}$ is always zero even $Cov(\mu,Z_i)$ is not always $0$.
My thoughts and work:
We know that the residual is $\hat{\mu_i}=Y_i-\hat{\beta_0}-\hat{\beta_1}X_i$ so...
$$\widehat{Cov(\hat{\mu},Z_i)}=\widehat{Cov(Y_i,Z_i)}-\widehat{Cov(\beta_0,Z_i)}-\widehat{Cov(B_1X_i,Z_i)}$$
and thus the second term would be equal to $0$ since $B_0$ doesn't have an $i$ term. But then I'm not sure how to proceed IF this is the right way to move forward.
Step by step help would be greatly appreciated!