What is the ruling of Islamic law regarding a wife threatening her husband with divorce due to his betrayal, and the husband saying, "Taliq (divorced) by three" or "Tariq (divorced) by three" with the intention of appeasing his wife, and does divorce occur in this case?

A wife is not permitted to seek a divorce without a legitimate Shar'i (Islamic legal) excuse, for the Prophet, peace and blessings be upon him, said: "Any woman who asks her husband for a divorce without any strong reason, the scent of Paradise will be forbidden to her." If marital infidelity means the husband marrying another woman without his wife's knowledge, then this is a legitimate right for the husband and is not considered infidelity or an excuse for divorce. However, if infidelity refers to committing adultery, then the wife should advise him and remind him of Allah. If he does not repent and insists on his action, then she is permitted to seek a divorce.

Divorce is in the hands of the husband, and he is to be believed regarding its non-occurrence. If the wife is certain that her husband pronounced "talaq" (divorce) three times, then, according to the majority of scholars, she becomes unlawful to him, and she must distance herself from him and not allow him to have conjugal relations with her. However, according to the school of thought of Sheikh al-Islam Ibn Taymiyyah, only one divorce occurs, and he can take her back before the completion of the 'iddah (waiting period), unless this divorce completes the third one. If the wife is not certain of what she heard, then there is no blame upon her for allowing him conjugal relations.