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 the neatherd Philoetius by showing him a scar on his thigh that he obtained during a boar hunt as a young man. This scar was known only to Odysseus and a few others, which convinced Philoetius of Odysseus's true 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.
Odysseus uses a scar on his leg as proof of his identity when he reveals himself to Penelope.
She makes Odysseus disguises as a beggar so no one will know his true identity.
Odysseus showed them his scar, which had been caused by a boar when he was a child. Only the real Odysseus would know about this scar, proving his identity to Philoetius and Eumaeus.
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 convinced his son, Telemachus, of his identity by revealing a secret detail about their home that only he would know. Additionally, Odysseus demonstrated his skill with the bow, which was a unique and recognizable characteristic of his. This combination of personal knowledge and a special skill convinced Telemachus of his father's true identity.
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.