Question

tan1/2=cscø-cotø

Answer 1

I will assume you are solving for ø
cscø-cotø = 1/2
1/sinø - cosø/sinø = 1/2
(1-cosø)/sin = 1/2
1 - cosø = sinø/2
square both sides
1 - 2cosø + cos^2 ø = sin^2 ø/4
4 - 8cosø + 4cos^2 ø = 1 - cos^2 ø
5cos^2 ø - 8cosø + 3 = 0
(cosø - 1)(5cosø - 3) = 0
cosø = 1 or cosø = 3/5
ø = 0, 360 or ø = appr 53.13° , 306.87°

but if ø = 0 or 360° , both cscø and cotø are undefined

furthermore, since I squared both sides, all answers must
be verified. ø=53.13 works, but ø = 306.87 gives me -1/2, not 1/2

so for 0 < ø < 360° , ø = 53.13°

Answer 2

Or, if you are trying to prove that
tan1/2 ø=cscø-cotø
start with
cos2ø = 2cos^2ø-1 = 1-2sin^2ø
Then
tan ø/2 = sin(ø/2)/cos(ø/2)
= √((1-cosø)/2) / √((1+cosø)/2)
= √(1-cosø)/√(1+cosø)
= √(1-cosø)^2 /√((1+cosø)(1-cosø))
= (1-cosø) / √(1-cos^2ø)
= (1-cosø) / √sin^2ø
= (1-cosø)/sinø
= cscø-cotø