Precalculus, Chapter 5, 5.4, Section 5.4, Problem 50

sin(u)=-7/25
using pythegorean identity,
sin^2(u)+cos^2(u)=1
(-7/25)^2+cos^2(u)=1
cos^2(u)=1-49/625=(625-49)/625=576/625
cos(u)=sqrt(576/625)=+-24/25
Since u is in quadrant III ,
:.cos(u)=-24/25
sin^2(v)+cos^2(v)=1
sin^2(v)+(-4/5)^2=1
sin^2(v)+16/25=1
sin^2(v)=1-16/25=(25-16)/25=9/25
sin(v)=sqrt(9/25)=+-3/5
since v is in quadrant III,
:.sin(v)=-3/5
cot(v-u)=cos(v-u)/sin(v-u)
cot(v-u)=(cos(v)cos(u)+sin(v)sin(u))/(sin(v)cos(u)-cos(v)sin(u))
plug in the values of sin(v),sin(u),cos(v) and cos(u),
cot(v-u)=((-4/5*-24/25+(-3/5)*-7/25))/((-3/5*-24/25-(-4/5)*-7/25))
cot(v-u)=(96/125+21/125)/(72/125-28/125)
cot(v-u)=(117/125)/(44/125)
cot(v-u)=117/44