prove the ff:

Question

a.) sin(x/2) - cos(x/2) = +-sqrt(1 - sin(x))

b.) tan(x/2) + cot(x/2) = 2csc(x)

Answer 1

let x/2 = Ø , then your equation becomes

sinØ - cosØ = ±√(1 - sin 2Ø)

we know that
if LS = RS , then
(LS)^2 = (RS)^2

LS = sinØ - cosØ
(LS)^2 = (sinØ - cosØ)^2
= sin^2 Ø - 2sinØcosØ + cos^2 Ø
= 1 - sin 2Ø
= (RS)^2 , could stop here, but carry on ....

LS = ± √(1 - sin 2Ø)
= RS

Answer 2

2nd part

I will make a similar substitution
let x/2 = y, then we have

prove: tany + coty = 2cscy

LS = siny/cosy + cosy/siny
= (sin^2 y + cos^2 y)/sinycosy
but (2sinycosy = siny)

= 1/((1/2)siny)
= 2cscy
= 2csc(x/2)
= RS