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))
