What is the ruling on my brother's wife remaining with her husband after discovering his secret marriage to another woman, given that the second wife is his friend's divorcee, and it is an unrecorded marriage known only to the two witnesses and the wife's family? And how can this situation be handled for fear of the matter being exposed to the friend?

Your brother's marriage to the divorced woman is valid if it takes place after her waiting period has ended, with the permission of her guardian, and in the presence of witnesses. Its designation as " عرفي" (customary) does not harm its validity, nor is it considered a secret marriage. Announcing the marriage is recommended but not obligatory. It is not necessary to inform the second wife of the marriage to another woman, and she is not entitled to request a divorce for this reason unless there is a legitimate شرعي justification. Your brother is advised to be just between his two wives, and the second wife is advised to fear Allah and not request a divorce without a legitimate reason.