Which method do biologists apply to identify and classify organisms based on their development and evolution over billions of years?

The most appropriate answer would be taxonomy.

Taxonomy is the branch of science that deals with identification and classification of different organisms.

It mainly classifies different organisms into 8 main taxa or groups. The principal ranks used in modern taxonomic hierarchy are

It used various principles to identify and classify an organisms such as cellular organisation, mode of nutrition, mode of reproduction, phylogentic or evolutionary relationships et cetera.

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