Answer
$sin\frac{x}{2}=\sqrt {\frac{3+2\sqrt 2}{6}}$
$cos\frac{x}{2}=\sqrt {\frac{3-2\sqrt 2}{6}}$
$tan\frac{x}{2}=3+2\sqrt 2$
Work Step by Step
Given $csc(x)=3, 90^\circ\lt x\lt 180^\circ$,
we have $sin(x)=\frac{1}{3}, cos(x)=-\frac{2\sqrt 2}{3}$ and $45^\circ\lt \frac{x}{2}\lt90^\circ$
$sin\frac{x}{2}=\sqrt {\frac{1-cos(x)}{2}}=\sqrt {\frac{3+2\sqrt 2}{6}}$
$cos\frac{x}{2}=\sqrt {\frac{1+cos(x)}{2}}=\sqrt {\frac{3-2\sqrt 2}{6}}$
$tan\frac{x}{2}=\frac{sin\frac{x}{2}}{cos\frac{x}{2}}=\sqrt {\frac{3+2\sqrt 2}{3-2\sqrt 2}}=\sqrt {\frac{(3+2\sqrt 2)(3+2\sqrt 2)}{(3-2\sqrt 2)(3+2\sqrt 2)}}=3+2\sqrt 2$
