Does a divorce that was genuinely intended while angry, and the wife actually did what the divorce was contingent upon during her menstrual period, take effect? And how can she be taken back without her family's knowledge?

If a man makes his wife's divorce contingent on an action, and that action occurs, the divorce takes effect if he intended divorce by it. Indeed, according to the majority of scholars, it takes effect even if he did not intend it. The anger of the divorcer does not prevent the divorce from taking effect, unless he was unconscious. Divorcing a menstruating woman is valid, even though it is an innovative (bid'i) divorce.

For reconciliation (raj'ah): If this divorce was not the third, and the wife is still within her waiting period (iddah), it is permissible to take her back without a new contract. Her consent or the consent of her guardian is not required, and having witnesses for the reconciliation is recommended. If her waiting period has ended, reconciliation is not permissible except with a new contract, and a valid offer and acceptance between the husband and the wife's guardian, in the presence of two just witnesses, is sufficient.