Before I prove your equation, I went through this and I think you meant #=1/tan((A-B)/2)# and not #-tan(A-B)/2# or #-tan((A-B)/2)#
So, you are trying to prove that #(cos A+cos B)/(sin A-sin B) = 1/tan((A-B)/2)#
#cos A+cos B=2cos((A+B)/2)cos((A-B)/2)#
#sinA-sinB=2sin((A-B)/2)cos((A+B)/2)#
#(cos A+cos B)/(sin A-sin B) = (cancel(2)cos((A-B)/2)cancel(cos((A+B)/2)))/(cancel(2)sin((A-B)/2)cancel(cos((A+B)/2)))#
#=cos((A-B)/2)/sin((A-B)/2)#
#=1/tan((A-B)/2)#
#=cot((A-B)/2)#
Proof:
#=1/tan((23-9)/2)=8.144346428#
#(cos 23+cos 9)/(sin 23-sin 9)=8.144346428#
