Given # tan (beta/2)=4tan (alpha/2)#
then #tan ((beta-alpha)/2)#
#=(tan (beta/2)-tan(alpha/2))/(1+tan(beta/2)tan(alpha/2))#
#=(4tan (alpha/2)-tan(alpha/2))/(1+4tan(alpha/2)tan(alpha/2))#
#=(3tan (alpha/2))/(1+4tan^2(alpha/2))#
#=((3sin (alpha/2))/cos(alpha/2))/(1+(4sin^2(alpha/2))/cos^2(alpha/2)#
#=(3*2sin (alpha/2)*cos(alpha/2))/(2cos^2(alpha/2)+8sin^2(alpha/2))#
#=(3sin alpha)/(1+cosalpha+4(1-cosalpha))#
#=(3sin alpha)/(5-3cos alpha)=RHS#