Is it permissible to seek divorce from a husband who refuses to declare his second marriage publicly for fear of problems, lies constantly, and causes his wife to lose her sense of security and trust in him, despite his kind treatment and provision of housing and expenses?

A husband is not obliged to inform his wife about his marriage to another woman. However, he must be just among his wives in terms of lodging, unless one of them waives her right. Lying is a blameworthy and forbidden characteristic, except in cases where the Sharīʿah has permitted it: reconciling disputes, in war, and a man speaking to his wife and a woman speaking to her husband. The intent behind a husband lying to his wife is to show affection and make promises that are not binding. However, deception to prevent a right or take what is not one's own is forbidden. If the husband lies in matters not permissible for him, the wife should advise him. If he persists in lying or fails to act justly in dividing his time, she has the right to seek ṭalāq (divorce) or khulʿ (divorce at the wife's instance with compensation), and she also has the right to seek ṭalāq due to infertility. However, ṭalāq is not a light matter and should not be resorted to unless reconciliation becomes impossible.