For all natural numbers, find the polynomials such that cos(n theta)=p_n(tan(theta))cos^n(theta)

Kasey Reese 2022-10-11 Answered
For all natural numbers, find the polynomials such that
cos ( n θ ) = p n ( tan ( θ ) ) cos n ( θ )
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Answers (1)

encaselatqr
Answered 2022-10-12 Author has 11 answers
Let q n ( x ) = ( 1 + i x ) n
Then
q n ( tan ( θ ) ) cos n ( θ ) = ( 1 + i sin θ cos θ ) n cos n θ = ( cos θ + i sin θ ) n = cos n θ + i sin n θ
So p n ( x ) = 1 2 ( q n ( x ) + q n ( x ) )
Use that cos ( x ) = cos x to show that
p n ( tan θ ) cos n θ = cos n θ
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