We consider an optical pulse propagating in a tapered photonic crystal fiber (PCF) wherein dispersion as well as nonlinearity varies along the propagation direction. The generalized nonlinear Schrödinger equation aptly models the pulse propagation in such a PCF. The design of the tapered PCF is based on the analytical results, which demand that the dispersion decrease exponentially and the nonlinearity increase exponentially. In this paper, we adopt the generalized projection operator method for deriving the pulse-parameter equations of the Lagrangian variation method and the collective variable method. Besides, we consider another pulse profile called raised cosine (RC), which is aimed at replacing the conventional hyperbolic secant pulse. From the detailed results, we infer that the initial RC pulse evolves into a hyperbolic secant pulse. Further, in order to minimize the input power requirement, we employ the idea of replacing the solid core in the PCF with chloroform. In addition to the single pulse compression, we also investigate the possibility of multisoliton pulse compression. Here, we consider eight chirped hyperbolic secant pulses as input and generate a train of ultrashort pulses at 850 nm based on the chirped multisoliton pulse compression. In a similar way, we extend this pulse compression with eight RC pulses.