If $\mathrm{Tan}A=\frac{-12}{5}$  and  ${630}^{\circ }<A<{720}^{\circ }$ Find the values of 

i) $\sin \frac{A}{2}$  ii) $\cos \frac{A}{2}$   iii) $\tan \frac{A}{2}$   iv) $\cot \frac{A}{2}$

$\tan A=\frac{-12}{5},{630}^{\circ }<A<{720}^{\circ }\Rightarrow A\in Q_4$

$\Rightarrow \frac{{630}^{\circ }}{2}<\frac{A}{2}<\frac{{720}^{\circ }}{2}$

$\Rightarrow {315}^0<\frac{A}{2}<{360}^{\circ }\Rightarrow \frac{A}{2}\in Q_4$

$\therefore$ $\cos A=\frac{5}{13}\therefore \cos A=\frac{5}{13}$$\left(\because A\in Q_4\right)$

i)  $\sin \frac{A}{2}=-\sqrt{\frac{1-\mathrm{Cos}A}{2}}=-\sqrt{\frac{1-\frac{5}{13}}{2}}$

$=-\sqrt{\frac{13-5}{26}}=-\sqrt{\frac{8}{26}}$    $\left(\because A\in Q_4\right)$

$=-\sqrt{\frac{13-5}{26}}=-\sqrt{\frac{8}{26}}=-\sqrt{\frac{4}{13}}=-\frac{2}{\sqrt{13}}$   $\left(\because A\in Q_4\right)$

ii) $\mathrm{Cos}\frac{A}{2}=\sqrt{\frac{1+\mathrm{Cos}A}{2}}$

$=\sqrt{\frac{1+\frac{5}{13}}{2}}=\sqrt{\frac{13+5}{26}}=\sqrt{\frac{18}{26}}$

$=\sqrt{\frac{9}{13}}=\frac{3}{\sqrt{13}}$

iii)  $\tan \frac{A}{2}= \frac{\sin {\displaystyle \raisebox{1ex}{$A$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}{\cos A/2}$

 $=\frac{{\displaystyle \raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$\sqrt{13}$}\right.}}{{\displaystyle \raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$\sqrt{13}$}\right.}}$

$=\frac{-2}{3}$

iv)  $cot \frac{A}{2}= \frac{1}{\tan {\displaystyle \frac{A}{2}}}$

$= \frac{1}{{\displaystyle \frac{-2}{3}}}$

$=\frac{-3}{2}$