Let α be a real number, not an odd multiple of π. Prove that tan α/2 is rational iff cos α and sin α are rational.
Solution
Put t = tan α/2. Then cos α = (1-t2)/(1+t2), sin α = 2t/(1+t2). So certainly cos α and sin &alphap are rational if t is.
Put k = cos α. Then since α is not an odd multiple of π, k is not -1. Hence t2 = (1-k)/(1+k), which is rational. Hence 2t = (1+t2) sin α is rational, and so also t.
© John Scholes
jscholes@kalva.demon.co.uk
29 Oct 2003
Last corrected/updated 29 Oct 03