Why is it important for Ulysses to keep his disguise in the scene where he is reunited with his father? Why does he feel the need to test his father’s loyalty in this way?

He wants to ensure that even his own family will be by his side regardless. He had nothing to do at that point but to see if his father also really knows him. Trust is also important in the family as much as it is important when meeting strangers and acquaintances

Explanation:

In his 20 years away from his home, Ulysses struggles through the Trojan War and a difficult journey home. He overcomes all the obstacles put in his path through his own wits and with the help of the gods. His lengthy struggle and the many betrayals seem to have affected him psychologically. He learns that he cannot trust people. He even approaches his son and wife while in disguise.

His father remains a suspect for disloyalty until Ulysses tests him. He tells Laertes a story about how his alter ego once gave shelter to Ulysses. He sees Laertes choke up and cry on hearing about his son, which proves his loyalty.