Heavy tailed distributions are worthwhile in modeling heavy tailed data. The researchers are often in search of such distributions to provide best fit to heavy tailed data. In this article, a new T-X family member called, a new exponential cosine-X family is introduced. A special sub-model of the proposed family, called, a new exponential cosine Weibull distribution is studied in detail. Some mathematical properties along with the useful series expansion of distribution and density functions of the proposed class are obtained. Two useful characterizations of this family are also provided. We consider the maximum likelihood and Bayesian estimation procedures to estimate the parameters of the proposed family. Monti Carlo simulation study is done to access the behavior of these estimators. For the illustrative purposes, a real-life application of the proposed family to a heavy tailed medical care insurance data set is provided. Finally, Bayesian analysis and performance of Gibbs sampling for the medical care insurance data are also carried out.