Odysseus is brave for many reasons. First, we know that he was brave because he's been in a very long, very bloody war and is making his way home. To have been in battle that long takes a certain amount of courage. Secondly, he bravely sets about the tasks that are set to him. He faces numerous monsters, including the Cyclops, and manages to defeat them. Lastly, he is brave enough to face what may or may not await him at home, despite his very long absense.
