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.
The derivation of basic identities can be done easily by using the functions of a right triangle. For easy reference, these trigonometric functions are listed below.
a/c = sin θ
b/c = cos θ
a/b = tan θ
c/a = csc θ
c/b = sec θ
b/a = cot θ