The detail from The Odyssey that best shows that Odysseus can be vain is that he risks unnecessary danger to his crew because he wants to taunt the cyclops.
The Odyssey is one of two most significant ancient Greek epic poems attributed to Homer. It is a sequel to the Iliad, the other Homeric epic.
The poem mainlydeals with the Greek hero Odysseus (called Ulysses in Roman myths), king of Ithaca, and his journey home after the fall of Troy.