[1-cosA+cosB-cos(A+B)]/1+cosA-cosB-cos(A+B)=tan A/2 cot B/2

take the numerator of L.H.S 

1-cosA+cosB-cos(A+B) simplifying 1-cosA = 2sin²  A/2 by half angle identity

                                               cosB-cos(A+B) = 2sin(A/2)sin((A+2B)/2)  difference into product identity

2sin²  A/2 + 2sin(A/2)sin((A+2B)/2)  by simple trigonometric identities  

by further simplifying

2sin(A/2)(sin A/2 + sin((A+2B)/2) 

now applying sum to product identity

2sin(A/2)(2sin((A+B)/2)cosB/2-------------1

and then take denominator

 1+cosA-cosB-cos(A+B)

 similarly by half angle and sum into product identity we get

2cos²  A/2 + 2cos(A/2)cos((A+2B)/2)

by further simplifying

2cos(A/2)(cos A/2 - cos((A+2B)/2)

 applying difference to product identity we get

2cos(A/2)(2sin((A+B)/2)sinB/2---------------2

now dividing 1 by 2

we get