Exercise 20, § 1

Let D be the subset of $\mathbf{C}$ which is the union of the set defined by $-\pi < \mathscr{R}(z) \leq \pi$, $\mathscr{I}(z) > 0$, and the segment $\mathscr{I}(z) = 0,\ 0 \leq \mathscr{R}(z) \leq \pi$. Show that the restriction of the function $\cos z$ to D is a bijection of D onto $\mathbf{C}$; the restriction of $\cos z$ to the interior of D is a homeomorphism of this open set onto the complement, in $\mathbf{C}$, of the half-line $y = 0,\ x \leq 1$.