The simplest proof of Jacobi's triple product identity originally due to Cauchy (1843) and Gauss (1866) is reviewed. In the same spirit, by means of induction principle and finite difference method, we prove a finite form of the celebrated quintuple product identity (Watson, 1929). Similarly, we give a new proof of an algebraic identity (Guo and Zeng, 2005) which can be considered as another finite form of the quintuple product identity.