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tan (a/2) is rational iff cos a and sin a are rational

Source: Eotvos 1906 p1

September 6, 2024
trigonometryalgebra

Problem Statement

Prove that, if tan⁡(a/2)\tan (a/2) is rational (or else, if a a is an odd multiple of π\pi so that tan⁡(a/2)\tan (a/2) is not defined), then cos⁡a\cos a and sin⁡a\sin a are rational. And, conversely, if cos⁡a\cos a and sin⁡a\sin a are rational, then tan⁡(a/2)\tan (a/2) is rational unless aa is an odd multiple of π\pi so that tan⁡(a/2)\tan (a/2) is not defined.
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