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Question

Prove that An=cosnθ if it is know that
A1=cosθ,A2=cos2θ
and for every natural number m > 2, the relations
Am=2A(m1)cosθAm2 hold.

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Solution

The statement is valid for n = 1 and n = 2
Now let Am2=cos(m2)θ,
Am1=cos(m1)θ
Then Am=2cosθAm1Am2
=2cosθcos(m1)θcos(m2)θ
=cosmθ+cos(m2)θcos(m2)θ
= cos mθ
It follows by mathematical induction that
An = cosnθ for all positive integers n.

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