Show that $\text{sin}(\bar z)$ is not holomorphic using uniqueness theorem. I want to show that $\text{sin}(\bar z)$ is not analytic using the uniqueness theorem.
The theorem essentially states that if we have a series $z_n$ such that non-constant $f(z_n)$ is zero for each $n$, then the function is not holomorphic if the infinite limit exists, but is not equal to any $z_n$.
The problem is $\text{sin}(\bar z)$ has zeros at $z=n\pi$. The theorem is directly of no help. What transform should be performed? 

If $\sin (\overline {z})$ is holomorphic then it must coincide everywhere with $\sin \, z$ because these two holomorphic functions are equal on the real line (which has limit points). This is a contradiction because these functions are not equal when $z=i$.
