Double Angle Formulas/Cosine/Proof 4

Proof


Consider an isosceles triangle $\triangle ABC$ with base $BC$, and apex $\angle BAC = 2 \alpha$.

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

From Angle Bisector and Altitude Coincide iff Triangle is Isosceles:
 * $AH \perp BC$

From Law of Cosines:

From Pythagoras's Theorem:

By definition of sine:

By definition of cosine:


 * $AH = AB \cos \alpha = AC \cos \alpha$

So:

Now:

Hence we get the equation: