as we know mass is costant.then why scientists searching for evidences to prove that inertial and gravitational masses are equivalent or same?

Mass is never constant... it varies with velocity.. this is practically observed.... thats the reason why..., SOMMERFIELD had to propose an atomic model other then BOHR'S model... since bohr considered that there is no relativistic change of mass... he proposed circular orbits for electron motion... but since that assumption was  wrong... sommerfield proved that the orbitals are elliptica and not circular... wherein it can be explained that the mass changes wrt to velocity..  even einstein in his relativity theory... proves that there is change in mass of the body with its velocity and hence he gives an equation relating variable mass n rest mass....

 Momentum is constant.

Mass is not.

and thrfr the research.

To keep momentum constant mass is taken as variable.

There are several theories based on this.