In by case method, we are trying to check the if the given function/propositional formula has the same behavior/truth values when we fix the truth value of one variable. If they have the same propositional formulas then they will have the same truth values. If we are checking for tautology in 2 variable then the formula would be represented by these 2 variables. So, reducing the formula to just one variable would help in checking in an efficient manner.