When does the waiting period (iddah) end for a wife divorced with a first repudiation (talaq) during a period of purity in which she was not touched, and is it necessary to obtain a divorce certificate after the period during which reconciliation is possible has ended?

The divorced wife observes an 'iddah (waiting period) of three menstrual cycles. The divorce takes effect as soon as it is pronounced. The husband has the right to take back his wife before her 'iddah expires, and the phrase "I have taken you back" is among the expressions used for reconciliation.