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cos^3 y / cos x + sin^3 y/ sin x =1 if cos y / cos x + sin y/ sin x =-1

Source: 1977 Swedish Mathematical Competition p4

March 26, 2021
algebratrigonometry

Problem Statement

Show that if cos⁡xcos⁡y+sin⁡xsin⁡y=−1 \frac{\cos x}{\cos y}+\frac{\sin x}{\sin y}=-1 then cos⁡3ycos⁡x+sin⁡3ysin⁡x=1 \frac{\cos^3 y}{\cos x}+\frac{\sin^3 y}{\sin x}=1
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