If you mean the test that Penelope set him (the rooted bed one), well then yes he was happy but not "completely happy" because he had a lot of loose ends to tie up (what with all the different suitors' families out to get him).
No one did- Penelope ran a series of tests to identify Odysseus.
Odysseus tells her so directly. Penelope then tests Odysseus, and he passes her test, confirming his identity.
The stranger tests Penelope's loyalty and patience by trying to woo her in Odysseus's absence. Penelope uses the stranger's presence as a way to buy time and delay having to choose a suitor, while secretly hoping for Odysseus's return.
The beggar tells Penelope a false story about meeting Odysseus and receiving news of his return. He claims to have met Odysseus and that he will soon return to reclaim his house and wife. Penelope later tests the beggar by asking for details about Odysseus, which the beggar fails to provide accurately.
Penelope believes that the beggar is her husband, Odysseus, who has returned home in disguise after being away for many years. She tests him by asking him to move their bed, a task only Odysseus could accomplish due to its special construction.
It had been twenty years so she had doubts.
Penelope tests Odysseus by asking him about their bed, which only the true Odysseus would know was built inside a living olive tree. This test confirms his identity to Penelope and proves that he is her husband returned after a long absence.
Odysseus' wife was Penelope, daughter of Icarius and Periboea.
Because she tests him with moving the huge bed in their chamber, which coincidentaly was made by Odysseus before his troubles. Odysseus therefore knows that the bed is made out of the huge olive tree that grows within the house.
he is the housekeeper for penelope and penelope is odysseus wife and odysseus is the king of ithleca
No, Penelope does not recognize that the beggar is Odysseus.
Penelope's first reaction to Odysseus in part 4 of Homer's "The Odyssey" is disbelief. She is hesitant to believe that the beggar in her presence is truly her husband returned after so many years, and she tests him to confirm his identity.