Half angle question (1 Viewer)

[alez]

Well, I seem to have totally forgotten how to use the half angle formulae
Can someone explain how to prove
(sin(x/2)-cos(x/2))^2 -1 = -sinx

 

[3unitz]

Well, I seem to have totally forgotten how to use the half angle formulae
Can someone explain how to prove
(sin(x/2)-cos(x/2))^2 -1 = -sinx

LHS = sin²(x/2) - 2 sin(x/2) cos(x/2) + cos²(x/2) - 1
= - 2 sin(x/2) cos(x/2) (red parts add to give 1)

we know: sin(2x) = 2 sinx cosx,
sub x = a/2
sin a = 2 sin (a/2) cos (a/2)

hence - 2 sin(x/2) cos(x/2) = - sinx = RHS