Double Angle Formulas/Cosine/Proof 4

Proof


Consider a Isosceles Triangle $\triangle ABC$ with base $BC$, and head angle $\angle BAC = 2 \alpha$.

Draw a angle bisector to $\angle BAC$ and name it $AH$.
 * $\angle BAH = \angle CAH = \alpha$

From Angler Bisector and Altitude coincide iff triangle is isosceles:
 * $AH \perp BC$.

From Law of Cosines:

From Pythagoras's Theorem:

By definition of sin:

By definition of cos:
 * $AH = AB \cos \alpha = AC \cos \alpha$

And so:

Now:

And so we get the equation: