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 $90^\circ\lt \frac{x}{2}\lt 135^\circ$
$\sin\frac{x}{2}=\sqrt {\frac{1-\cos(x)}{2}}=\sqrt {\frac{26+5\sqrt {26}}{52}}$
$\cos\frac{x}{2}=-\sqrt {\frac{1+\cos(x)}{2}}=-\sqrt {\frac{26-5\sqrt {26}}{52}}$
$\tan\frac{x}{2}=\frac{\sin\frac{x}{2}}{\cos\frac{x}{2}}=-\sqrt {\frac{26+5\sqrt {26}}{26-5\sqrt {26}}}=-\sqrt {\frac{(26+5\sqrt {26})(26+5\sqrt {26})}{(26-5\sqrt {26})(26+5\sqrt {26})}}=-5-\sqrt {26}$
