Odysseus proves his identity to different people in different ways:
He shows the swineherd Eumaeus and the stockman Philoetius his boar scar.
He strings his bow and shoots the arrow through the handles of the 12 axes.
He tells Penelope that the bed cannot be moved.
He tells his father Laertes each of the crops that were planted from year to year.
She makes Odysseus disguises as a beggar so no one will know his true identity.
He revealed his true identity because he was boasting about being clever and stuff. He wasn't being very bright, as usual.
The right way
Odysseus is treated as a guest by the Phaeacians because he didn't want them to find out his true identity or else they would have killed him.
his identity
Odysseus proved his identity to Philoetius and Eumaeus by revealing a scar on his thigh that was left from a hunting accident many years ago. This scar was known only to them and Odysseus, thus confirming his true identity.
Odysseus proves his identity to the cowherd and the swineherd by revealing a scar on his thigh, which was inflicted by a boar when he was young. This scar is a unique and personal detail that only Odysseus would know about, confirming his true identity to the cowherd and the swineherd.
She makes Odysseus disguises as a beggar so no one will know his true identity.
Odysseus uses a scar on his leg as proof of his identity when he reveals himself to Penelope.
He revealed his true identity because he was boasting about being clever and stuff. He wasn't being very bright, as usual.
The right way
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 proves his identity to Penelope by revealing a secret detail about their bed, which only he would know. He explains that he built their bed from an olive tree trunk still rooted in their home, making it impossible to move. This intimate detail convinces Penelope of his true identity as her husband.
Odysseus is treated as a guest by the Phaeacians because he didn't want them to find out his true identity or else they would have killed him.
how did odysseus prove who he was
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.
Eumaeus has proven his faithfulness to Odysseus by taking care of his household and showing loyalty to his master. In previous books, he has remained true to Odysseus despite the suitors' presence. Eumaeus has also helped Odysseus, disguised as a beggar, by offering him food and shelter without knowing his true identity.