First we can calculate #sin(x/2)# and #cos(x/2)# using the double angle formula:
#cosx=cos^2(x/2)-sin^2(x/2)=2cos^2x-1#
So we can calculate the cosine by solving:
#2cos^2(x/2)-1=-4/5#
#2cos^2(x/2)=1/5#
#cos^2(x/2)=1/10#
#cos(x/2)=-sqrt(10)/10 vv cos(x/2)=sqrt(10)/10#
If #pi < x< 3/2pi# then #pi/2< x/2 < 3/4pi#, so #x/2# lies in the 2nd quadrant, so only sine is positive.
#cos(x/2)=-sqrt(10)/10#
Sine can be calculated using:
#sin^2(x/2)+cos^2(x/2)=1#
#sin^2(x/2)=1-cos^2(x/2)=1-1/10=9/10#
#sin(x/2)=sqrt(9/10)=(3sqrt(10))/10#
Now we can calculate #tan(x/2)# and #cot(x/2)#
#tan(x/2)=sin(x/2)/cos(x/2)=((3sqrt(10))/10)-:(-sqrt(10)/10) #
#tan(x/2)=((3sqrt(10))/10)xx(-10/sqrt(10))=-3#
#cot(x/2)=1/tan(x/2)=-1/3#
