Am I entitled to claim the price difference between the appraisal value of the house determined by his brother and the actual price at which he sold the house eight months later?

If the brother and sister agreed that he would compensate her for her share of the house according to a specific valuation, and she accepted that, then she is only entitled to the agreed-upon amount, whether it is less or more than the price at which the house was later sold. This is because it is considered a sale or a gift. However, if the sister did not agree with her brother and did not accept the valuation, then her right to the house remains. She is entitled to half of the selling price, whatever it may be, and she has the right to demand half of the rent for the period during which the brother resided there, unless she waives her right.