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Trig Polynomial

Source: INMO 2020 P2

January 19, 2020
algebrapolynomialtrigonometryINMO 2020

Problem Statement

Suppose P(x)P(x) is a polynomial with real coefficients, satisfying the condition P(cos⁡θ+sin⁡θ)=P(cos⁡θ−sin⁡θ)P(\cos \theta+\sin \theta)=P(\cos \theta-\sin \theta), for every real θ\theta. Prove that P(x)P(x) can be expressed in the formP(x)=a0+a1(1−x2)2+a2(1−x2)4+⋯+an(1−x2)2nP(x)=a_0+a_1(1-x^2)^2+a_2(1-x^2)^4+\dots+a_n(1-x^2)^{2n}for some real numbers a0,a1,…,ana_0, a_1, \dots, a_n and non-negative integer nn.
Proposed by C.R. Pranesacher
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