Why is the Half-Angle formula derived from the Double-Angle formula?

Isn't that like ending back to where you started? Half a double?

Obviously not or this wouldn't work! So can someone explain to a dummy like me? :)

Question

Why is the Half-Angle formula derived from the Double-Angle formula?

Isn't that like ending back to where you started? Half a double?

Obviously not or this wouldn't work! So can someone explain to a dummy like me? :)

anonymous · Answer

gimmick is to take $\cos(2x)$ formula and replace $2x$ by $x$ and $x$ by $\frac{x}{2}$ then you solve for $\cos(\frac{x}{2})$  in the resulting equation

anonymous · Answer

yes, it is a gimmick, no, you do not get back what you started because you are solving for $\cos(\frac{x}{2})$

anonymous · Answer

i know how to get the formula, I just don't get the why? haha :)