Question

Given the following: sin(z) = (eiz - e−iz)/2i cos(z) = (eiz + e−iz)/2 sin(z)cos(w)−cos(z)sin(w)=sin(z−w) sin(z)=0 has solution z=kπ for some integer k tan(z)=tan(w) has solution z=w+kπ for some integer k The natural domain of the function z↦tan(z) is D = {z∈C|z≠(2m+1)π/2, m∈ Z} Question: Use the above to find and sketch a subset U of D so that: the function f:U↦C given by f(z)=tan(z) is injective AND the set of real numbers {|z||z∈U} is not bounded above AND the set of real numbers {arg(z)|z∈U} is equal to [0, π].