Answer
$sin\frac{x}{2}=\sqrt {\frac{26+5\sqrt {26}}{52}}$
$cos\frac{x}{2}=-\sqrt {\frac{26-5\sqrt {26}}{52}}$
$tan\frac{x}{2}=-26-5\sqrt {26}$
Work Step by Step
Given $cot(x)=5, 180^\circ\lt x\lt 270^\circ$,
we have $tan(x)=\frac{1}{5}, sin(x)=-\frac{\sqrt {26}}{26}, cos(x)=-\frac{5\sqrt {26}}{26}$ and $90^\circ\lt \frac{x}{2}\lt 135^\circ$
$sin\frac{x}{2}=\sqrt {\frac{1-cos(x)}{2}}=\sqrt {\frac{26+5\sqrt {26}}{52}}$
$cos\frac{x}{2}=-\sqrt {\frac{1+cos(x)}{2}}=-\sqrt {\frac{26-5\sqrt {26}}{52}}$
$tan\frac{x}{2}=\frac{sin\frac{x}{2}}{cos\frac{x}{2}}=-\sqrt {\frac{26+5\sqrt {26}}{26-5\sqrt {26}}}=-\sqrt {\frac{(26+5\sqrt {26})(26+5\sqrt {26})}{(26-5\sqrt {26})(26+5\sqrt {26})}}=-26-5\sqrt {26}$
