MathDB

1

Part of 1906 Eotvos Mathematical Competition

Problems(1)

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

Source: Eotvos 1906 p1

9/6/2024
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.
trigonometryalgebra