The relevant trig identities for half angles are as follows:

sin(x) = 2 tan(x/2)/(1+tan²(x/2))
cos(x) = (1-tan²(x/2))/(1+tan²(x/2)

Also, quotients formed from rational numerators and denominators are also rational.

Since we are given that tan(x/2) is rational, the numerators and denominators of the above relations are rational, and hence the right-hand sides are rational. Therefore sin(x) and cos(x) are rational. Because tan(x)= sin(x)/cos(x), sec(x)=1/cos(x) and cosec(x)=1/sin(x), it follows that the six common trig functions are all rational.



Solution submitted by Donnie