When Odysseus disguises himself as an old man in disguise why does he need to keep his secret from his wife in The Odyssey?
The section I think you must be refering to is Book 19 of this excellent epic classic, when Odysseus and Penelope finally are able to meet by themselves. What is interesting about this meeting is that Odysseus continues to stay in disguise, tricking his wife into believing he is not who he is and also giving her news of her husband. Of course, the question we all have is why on earth at this stage in his adventure does Odysseus need to deceive his wife? They are alone and nobody else is around, so he could reveal his identity easily.
However, there are three possible answers we can give. The first lies in the nature of the character of Odysseus. Throughout the story, he is a character who is presented as being in love with deception. This is something that even Athene notices and praises him for. He is the kind of character that would rather deceive and show his cunning than tell the truth. Note how Athene had to command him to reveal himself to his son, Telemachus. The suggestion is that Odysseus would not have done so if it were left up to him.
Secondly, I think there is a sense in which Odysseus is very aware that he has been away for so long and his beautiful wife has been surrounded by many suitors. He wants to ensure her faithfulness to him but also see how she has changed.
Finally, Odysseus definitely wants to take advantage of his disguise to know what situation he is going to have to face in the inevitable showdown. He does not want to face the suitors without knowing more about them, and perhaps Penelope might show a change in her personality if she knew that her husband was actually there. Much better then to deceive her until a time when he is ready to challenge the suitors.