Odysseus reveals his true identity to Polyphemus because he is absorbed in the moment. He has just designed an ingenious escape and blinded a cyclops, and, caught up in the glory of the moment doesn't think about the consequenses of his actions. Hubris (excessive pride) is one of his character flaws.
He revealed his true identity because he was boasting about being clever and stuff. He wasn't being very bright, as usual.
He did not say anything to Poseidon; rather, he blinded Polyphemus, the god's cyclops son, because of his pride (his hubris) Odysseus revealed his true identity to the cyclops, and in return Polyphemus told his father (the resulting hamartia.)
He tells her about their bed and how its made out of the olive tree and immovable. Only something that Odysseus and her knew.
Odysseus escapes Polyphemus's cave by employing cunning and trickery. He tells the Cyclops that his name is "Nobody," so when he blinds Polyphemus with a sharpened stake, the giant's cries for help go unheard, as he claims that "Nobody" is attacking him. To flee the cave, Odysseus and his men hide under the sheep as Polyphemus lets them out to graze, allowing them to escape unnoticed. Once they are safely at sea, Odysseus reveals his true identity, provoking Polyphemus's wrath.
Odysseus and his men then took the timber and heated the sharpened end in the fire until it glowed red. Then, with all their strength, they pushed the red-hot point into the eye of Polyphemus. The Cyclops howled and woke up flailing, but he was now blind.
Odysseus' decision to stay at Polyphemus' cave proves disastrous. When the Cyclops returns, he kills several of Odysseus' men and eats them. Odysseus manages to escape, but his blinding of Polyphemus, who was a son of Poseidon, has terrible consequences. When Odysseus finally reveals his true identity to Polyphemus, the monster prays to his father for vengeance. This prayer brings about Poseidon's persecution of Odysseus and in Odysseus' return from Troy being significantly delayed.
In chapter 9 of the Odyssey, Odysseus foolishly reveals his true identity to Polyphemus the Cyclops after blinding him. This action leads to Polyphemus seeking revenge and making it harder for Odysseus and his men to escape.
Odysseus does not reveal his true identity to his wife Penelope when he first returns to Ithaca because he needs to test her loyalty and the loyalty of his household members. He also wants to ensure that he has a strategic advantage in dealing with the suitors who have taken over his home in his absence. Additionally, revealing his true identity prematurely could jeopardize his plans for revenge and reclaiming his kingdom.
Not knowing the parentage of Polyphemus (his father was Poseidon, who was already upset with Odysseus), and being a proud man, he probably thought that there was nothing that Polyphemus could do to him once he was on his boat. He was wrong, of course.
When his fellow cyclops came into the cave, Polyphemus said that "Nobody" was trying to kill him (Nobody = a name). The others heard it as "nobody" was trying to kill him, meaning that he was not in danger (nobody = no one).
Odysseus' plan to blind Polyphemus using a sharpened stake was clever, as it allowed them to escape from the cyclops' cave by hiding under the bellies of his sheep. However, his decision to reveal his true name to Polyphemus after successfully escaping ultimately led to Poseidon's wrath and caused further challenges for Odysseus and his crew during their journey home.
In Book 9 of the Odyssey, Odysseus exemplifies his trickster nature through several key actions. First, he cleverly introduces himself to the Cyclops Polyphemus as "Nobody," which allows him to escape after blinding the giant. Second, he devises a cunning plan to intoxicate Polyphemus with wine, further disorienting him. Lastly, after escaping, Odysseus taunts Polyphemus by revealing his true identity, showcasing both his cleverness and his hubris.