a) Prove that $|z^i|< e^\pi$ for all complex $z \neq 0$.
b) Prove that there is no constant $M > 0$ such that $|\cos z| < M$ for all complex $z$.
Where complex powers are defined as $z^w=e^{w Log\ z}$ and complex cosine is $\cos z=\frac{e^{iz}+e^{-iz}}2$.
On
a) Since you define $z^i$ as $e^{i\log\lvert z\rvert-\operatorname{arg}z}$, where $\operatorname{arg}z$ is the only argument of $z$ from $[-\pi,\pi)$, then$$\lvert z^i\rvert=\lvert e^{i\log\lvert z\rvert-\operatorname{arg}z}\rvert=e^{-\operatorname{arg}z}<e^\pi.$$ b) Use the fact that $\cos(iz)=\cosh z$.
For the first question you have that $$ \left| w \right| = \left| {e^{ - \theta + i\log \left| z \right|} } \right| = e^{ - \theta } \left| {e^{i\log \left| z \right|} } \right| $$ but for any real $\alpha$ it is $$ \left| {e^{i\alpha } } \right| = 1 $$ thus $$ \left| w \right| = e^{ - \theta } \leqslant e^\pi $$ For the second question I thin you mean $|\cos z|$ instead $\cos z$. In this case you can use the Liouville Theorem: every entire bounded function is constant. Therefore, since $\cos z$ is entire and not constant it can not be bounded. In a more elementary way: if you choose $z=iy$ you have $$ \left| {\cos iy} \right| = \frac{1} {2}\left| {e^{ - y} + e^y } \right| $$ which of course is not bounded.