tangent function determines a local homeomorphism The question ask me to show that tangent function determines a local homeomorphism $\tan: \mathbb C \to \mathbb C P^1$. I don't understand what the question asking, is the question asking me to show that $\tan:\mathbb C \to \mathbb CP^1$ is a local homeomorphism?
 A: The question is asking you to show that the tangent function $\operatorname{tan}(z)\colon \mathbb{C}\rightarrow \mathbb{CP}^1$ is a local homeomorphism, I guess.
Let me assume that you defined the tangent function as the holomorphic mapping $\mathbb{C}\rightarrow \mathbb{CP}^1$associated to the meromorphic function $\frac{\operatorname{sin}}{\operatorname{cos}}\colon \mathbb{C}\rightarrow \mathbb{C}$. Here, $\operatorname{sin}(z)=\frac{e^{iz}-e^{-iz}}{2i}$ and $\operatorname{cos}(z)=\frac{e^{iz}+e^{-iz}}{2}$ denote the complex sine and cosine functions, respectively. In other words, the tangent function is the holomorphic mapping associated to the meromorphic function $\mathbb{C}\rightarrow\mathbb{C}; z \mapsto\frac{e^{2iz}-1}{i(e^{2iz}+1)}$.
Observe that the tangent function can be written as the composition of a linear automorphism of $\mathbb{C}$, the exponential function $\mathbb{C}\rightarrow \mathbb{C}\setminus\{0\}$, and a Möbius transformation. Namely, consider the maps $\phi\colon \mathbb{C}\rightarrow \mathbb{C};z\mapsto 2\pi i z$ and $\psi\colon\mathbb{C}\setminus\{0\}\rightarrow \mathbb{CP}^1; z\mapsto \frac{z-1}{iz+i}$. For all $z\in \mathbb{C}$, we then have $(\psi\circ {\operatorname{exp} }\circ \phi) (z)=\operatorname{tan}(z).$
The exponential function is a local homeomorphism by the inverse function theorem for holomorphic functions. From $\operatorname{det} \big(\begin{smallmatrix}
  1 & -1\\ 
  i & i
\end{smallmatrix}\big)=2i\neq0$, we see that the linear fractional transformation $\psi$ is bijective, thus a local homeomorphism. Hence, as a composition of local homeomorphisms, the tangent function is a local homeomorphism.

One can even show that the map $\operatorname{tan}\colon\mathbb{C}\rightarrow \mathbb{CP}^1\setminus\{\pm i\}$ is a covering map.
