Trigonometry problem - prove that $\cos(\beta)=1-1/(8\cos^2(\alpha))$

Question

$ABC$ is Isosceles triangle ($AB=AC$) and $\angle ABC= \angle ACB = \alpha$. A point $F$ is chosen on the opposite side of $AC$ from $B$ such that $AF=CF=BC$ and $\angle FAC = \angle FCA = \beta$. Show that $$\cos(\beta)=1-\frac{1}{8\cos^2(\alpha)}$$

I showed that if the question was true then $AB=AC=BC$ but I don't know how to prove the question..

Answer 1

$$BC=2AC\cos\alpha=2(2FC\cos\beta)=4BC\cos\alpha\cos\beta,$$ which gives $$4\cos\alpha\cos\beta=1.$$