sin x=1/9. We need the ratio for cos x so we can calculate the (x/2). So since we know that the opposite is 1 and the hypotenuse is 9 and we are in quadrant 1 from the given domain we can calculate the adjacent side using pythagorean theorem.
That is,
a=sqrt(9^2 -1^2)=sqrt(81-1)=sqrt80 = 4sqrt5. Hence,
cos x = (4sqrt5)/9, 0 < x< pi/2. We also need the domain for x/2 and we can get that by halving the given domain 0 < x/2 < pi/4 so x/2 is in Quadrant I. Since it is in quadrant I sin (x/2), cos(x/2) and tan(x/2) will all have positive answers.
Now lets calculate sin (x/2), cos(x/2) and tan(x/2),
sin (x/2)=sqrt(1/2 (1-cosx)
=sqrt(1/2 (1-(4sqrt5)/9)
=sqrt((9-4sqrt5)/18
=sqrt(162-72sqrt5)/18
cos (x/2)=sqrt(1/2 (1+cosx))
=sqrt(1/2(1+(4sqrt5)/9)
=sqrt((9+4sqrt5)/18
=sqrt(162+72sqrt5)/18
tan (x/2)=sinx/(1+cosx)
=(1/9)/(1+(4sqrt5)/9
=(1/9)/((9+4sqrt5)/9)
=1/cancel9 * cancel9/(9+4sqrt5)
=1/(9+4sqrt5) *(9-4sqrt5)/(9-4sqrt5)
=(9-4sqrt5)/(81-80
=9-4sqrt5