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values of . This suggests we examine the statement specifically for natural values of , and induction is the readiest tool.

Proposition. For any and , .

Proof. Fix an arbitrary real number , and let be the statement . We induct on .

Base case: The calculation verifies .

Induction step: We show the implication for any natural number . Assume the induction hypothesis: for a given value , the single case is true. Using the angle addition formula

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