The linear stability analysis of a pulsed flow in a linear Maxwell fluid confined in the Taylor-Couette system is investigated. Both cylinders are subjected to modulated rotation in phase with equal amplitude and frequency. We show that in the limit of low frequency, the Deborah number has no effect on the stability of the basic state which tends to a stable configuration. The basic state is potentially unstable at an intermediate frequency and it becomes more unstable as Deborah number increases. At high frequency limit, the Deborah number has a strong destabilizing effect. These numerical results are in good agreement with the asymptotic solutions obtained in the limit of high and low frequencies.