Sec b cot b =

sec b
csc b
cot b
tan b

Given: 
tan(B/2) = sec(B) / (sec(B) * csc(B) + csc(B)) 

Apply the half angle formula to convert tan(B/2) to terms of B: 
sin(B) / (1+cos(B)) = sec(B) / (sec(B) * csc(B) + csc(B)) 

Convert everything else to be in terms of sin and cos: 
sin(B) / (1+cos(B) = (1/cos(B)) / ((1/cos(B)) * (1/sin(B)) + (1/sin(B))) 

Multiply right side by "sin(B)/sin(B)" to simplify the fractions: 
sin(B) / (1+cos(B) = (sin(B)/cos(B)) / ((1/cos(B)) + 1) 

Change "1" to cos(B)/cos(B) and then combine over 
common denominator: 
sin(B) / (1+cos(B) = (sin(B)/cos(B)) / ((1/cos(B)) + cos(B)/cos(B)) 
sin(B) / (1+cos(B) = (sin(B)/cos(B)) / ((1+cos(B))/cos(B)) 

Dividing by a fraction equals multiplying by its reciprocal: 
sin(B) / (1+cos(B) = (sin(B)/cos(B)) * (cos(B) / (1+cos(B))) 

Multiply terms on the right side (canceling cos(B) terms): 
sin(B) / (1+cos(B) = sin(B) / (1+cos(B))