Derivation of the Half Angle Formulas

Half angle formulas can be derived from the double angle formulas, particularly, the cosine of double angle. For easy reference, the cosines of double angle are listed below:
 

cos 2θ = 1 - 2sin2 θ   →   Equation (1)
cos 2θ = 2cos2 θ - 1   →   Equation (2)

 

Note that the equations above are identities, meaning, the equations are true for any value of the variable θ. The key on the derivation is to substitute θ with ½ θ.
 

For Equations (1) and (2), let θ = ½ θ
 

From Equation (1)
cos2(12θ)=12sin212θ

cosθ=12sin212θ

2sin212θ=1cosθ

sin12θ=1cosθ2

 

From Equation (2)
cos2(12θ)=2cos212θ1

cosθ=2cos212θ1

2cos212θ=1+cosθ

cos12θ=1+cosθ2

 

tan12θ=sin12θcos12θ

tan12θ=1cosθ21+cosθ2

tan12θ=1cosθ1+cosθ   →   Equation (3)

 

From Equation (3)
tan12θ=1cosθ1+cosθ×1cosθ1cosθ

tan12θ=(1cosθ)21cos2θ

tan12θ=(1cosθ)2sin2θ

tan12θ=1cosθsinθ

 

From Equation (3)
tan12θ=1cosθ1+cosθ×1+cosθ1+cosθ

tan12θ=1cos2θ(1+cosθ)2

tan12θ=sin2θ(1+cosθ)2

tan12θ=sinθ1+cosθ