This is how I started:

a2 = (Zhat'*Zhat)^-1*Zhat'*y

= ((HZ)'(HZ))^-1*(HZ)'*y

= (Z'H'HZ)^-1*Z'H'y

= (Z'HZ)^-1*Z'Hy

= (Z'HZ)^-1*Z'Xb

= (Z'X(X'X)^-1*X'Z)^-1*Z'Xb

= (X'Z)^-1*(X'X)*(Z'X)^-1*Z'Xb

= (X'Z)^-1*(X'X)b

Can I simplify this further?

Thanks!!