Hope you know the identity
cos2A=(cosA)^2-(sinA)^2 which is equal to
cos2A=[(cosA)^2-(sinA)^2]/1
but (cosA)^2+(sinA)^2=1 therfor
cos2A=[(cosA)^2-(sinA)^2]/[(cosA)^2+(sinA)^2]
now dividing both numerator and denominator by (cosA)^2
we get
cos2A=[(1-(tanA)^2]/[(1+(tanA)^2]
now put A/2 in place of A and you get
cosA=[(1-(tanA/2)^2]/[(1+(tanA/2)^2]
but tanA/2=z therfor
cosA=[(1-z^2]/[(1+z^2]
hence proved