For $ N \ge 1$, consider the square contour $S_N$ with vertices at $\pi( N+1/2) + i\pi(N+1/2)$; $-\pi( N+1/2) + i\pi(N+1/2)$; $-\pi( N+1/2) - i\pi(N+1/2)$; and $\pi( N+1/2) - i\pi(N+1/2)$. This is a square with side ($2N+1)\pi$ cantered at the origin.
(a) Show that $|\sin(z)| \ge 1$ on the vertical sides and $|\sin(z)| \ge \sinh(\pi/2)$ on the horizontal sides.
(b) Use your solution in (a) to show that $$\lim_{N\rightarrow\infty}\left|\int_{S_N} \frac{1}{z^{2} \sin(z)}dz\right| = 0$$
I've found a similar solution to proving $|\sin(z)| \ge 1$ on the vertical sides, and I am able to understand the solution, though I am unsure of how to approach the horizontal sides. I've gotten up to $\sqrt{\cosh^{2}(y) - \cos^{2}(x)} \ge \sinh(\pi/2)$ but don't know where to go from here. Also just have no idea how to approach (b).
I just did the first one, it seems long but I just took the parametrization of the lines and then plug them into sin(z), use the identities Felipe gave on class (sinxcosiy+siniycosx etc) and also the definition of sinz (e^iz etc), I don't know but I assumed that N is integer and then it reduces to an easy function, changing billions of variables then find maximum by derivatives which gave out |sinz|>=1..Now onto the 2nd one, it looks ok, just do squeeze theorem from the bounds found in the first one.