You're told 'x' is in Q III then x/2 will be in Q II and be negative also
using the identity cos x/2 = +- sqrt ( (1+(-2/3)) /2) = +- sqrt 1/6 = +- 1/sqrt6 = sqrt 6 / 6
BUT it is in Q II - sqrt 6/6
I'll take a shot at this.
we now that cos x = -2/3
so cos x/2 = -2/3 x 1/2=-2/6=-1/3
Maybe -1/3?