Let $f$ be differentiable inside and on the unit circle, centered at the origin, traversed anticlockwise. Show that for $0<|z|<1$ $$2\pi i f(z) = \int_C \frac{f(\zeta)}{\zeta-z} \, d\zeta - \int_C \frac{f(\zeta)}{\zeta -\frac{1}{\bar z}} \, d\zeta$$
I know that the first of the two integrals equals the LHS by Cauchy's integral formula, and so the second must be zero. I know that Cauchy's theorem says that if the second function is holomorphic on some simply connected region containing $C$ then the integral is $0$. I think this means I need to show that $|1/\bar z|\geq1$. How do I do this?