Remember that 2x=x+x
sin(2x)=sin(x+x)=sinxcosx+cosxsinx=color (red) (2sinxcosx)
cos(2x)=cos(x+x)=cosxcosx-sinxsinx=color (red) (cos^2x-sin^2x
Now let 2x=theta so x=theta/2
costheta=cos^2(theta/2)-sin^2(theta/2)
costheta=2cos^2(theta/2)-1
2cos^2(theta/2)=costheta+1
cos^2(theta/2)=1/2(costheta+1)
color (red) (cos(theta/2)=+-sqrt(1/2(costheta+1)
Similarly,
costheta=1-2sin^2(theta/2)
2sin^2(theta/2)=1-costheta
sin^2(theta/2)=1/2(1-costheta)
color (red) (sin(theta/2)=+-sqrt(1/2(1-costheta)
tan(2x)=tan(x+x)=(tanx+tanx)/(1-tan^2x)=color (red) ((2tanx)/(1-tan^2x)
Finally tan(theta/2)=sin(theta/2)/cos(theta/2)
tan(theta/2)=(+-sqrt(1/2(1-costheta)))/(+-sqrt(1/2(costheta+1))
tan(theta/2)=sqrt(1-costheta)/sqrt(1+costheta)
tan(theta/2)=sqrt((1-costheta)(1+costheta))/(1+costheta)^2
color (red) (tan(theta/2)=sintheta/(1+costheta)