True/False: In general, a firm's profit is the same regardless of whether it solves a profit maximization or cost minimization problem.

Answer: True. Cost minimization is the "dual problem" of profit maximization. In other words, the maximum profits are the same, whether you use a cost minimization or profit maximization problem.
A detailed proof is probably beyond the scope of most undergraduate courses, but can be found in Mas-Colell, Whinston and Green.

(The output of a cost minimizing and profit maximizing firm need not be the same; consider the case of constant returns to scale.)