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cos 2x = cos 2y if sin x + cos y = 1 and cos x + sin y = -1

Source: 2005 Estonia National Olympiad Final Round grade 11 p1

April 13, 2020
trigonometrysystem of equationsalgebra

Problem Statement

Real numbers xx and yy satisfy the system of equalities {sin⁡x+cos⁡y=1cos⁡x+sin⁡y=−1\begin{cases} \sin x + \cos y = 1 \\ \cos x + \sin y = -1 \end{cases} Prove that cos⁡2x=cos⁡2y\cos 2x = \cos 2y.
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