Is it possible to determine the genotype of a person showing a dominant phenotype? A recessive phenotype? why?

For an individual trait, it is not possible to determine the genotype of a person who shows the dominant phenotype, because both zygous (e.g., BB) and heterozygous individuals (Bb) would show that trait. Only a zygous recessive individual (bb) would show the recessive trait, so that person would be the only one you could know the genotype of.