I see three possibilities I would consider a person who is addicted to prescription drugs to be a drug addict. I would consider a person who is addicted to prescription drugs as a drug addict. I would consider a person who is addicted to prescription drugs (a drug addict) / an addict.
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New2grammarWhich choices are correct?All the following are correct. (My symbolization is a little different from yours, but includes the same three possibilities. Parenthesization indicates "optional". )