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 = 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)=1−2sin212θ
cosθ=1−2sin212θ
2sin212θ=1−cosθ
From Equation (2)
cos2(12θ)=2cos212θ−1
cosθ=2cos212θ−1
2cos212θ=1+cosθ
tan12θ=sin12θcos12θ
tan12θ=√1−cosθ2√1+cosθ2
From Equation (3)
tan12θ=√1−cosθ1+cosθ×1−cosθ1−cosθ
tan12θ=√(1−cosθ)21−cos2θ
tan12θ=√(1−cosθ)2sin2θ
From Equation (3)
tan12θ=√1−cosθ1+cosθ×1+cosθ1+cosθ
tan12θ=√1−cos2θ(1+cosθ)2
tan12θ=√sin2θ(1+cosθ)2