Question

How do you prove that `sin^2(x/2) = (sin^2x)/(2(1+cosx))`?

Answer 1

We know that

`cos2x=cos^2x-sin^2x=1-2sin^2x`

Hence

`sin^2x=1/2*(1-cos2x)`

Set `x->t/2` so we have

`sin^2(t/2)=1/2*(1-cost)`

Multiply and divide the LHS with `1+cost` hence

#sin^2(t/2)=1/2*[((1-cost)*(1+cost))/(1+cost)]=> sin^2(t/2)=1/2*[(1-cos^2t)/(1+cost)]=> sin^2(t/2)=1/2*[sin^2t/(1+cost)]#

Which proves the requested