Prove that $ \begin{vmatrix} 1+x & 1 & 1 \\ 1 & 1+y & 1 \\ 1 & 1 & 1+z \end{vmatrix} = xy+yz+zx+xyz = xyz\ (1+\frac{1}{x}+\frac{1}{y}+\frac{1}{z}) $