This paper presents a method for fault-tolerant broadcasting in faulty hypercubes using a new metric called local safety. A new concept called broadcast subcube is introduced, according to which various techniques are proposed to improve performance of the broadcast algorithm. An unsafe hypercube can be split into a set of maximal safe subcubes. We show that if these maximal safe subcubes meet certain requirements given in the paper, broadcasting can still be carried out successfully and in some cases optimal broadcast is still possible. The sufficient condition for optimal broadcasting is also presented. Extensive simulation results are presented.