For a function:
$$f(x)=\frac{(1-x^2)}{(1+x^2)-2 x \cos \omega}$$
Let $x=-\frac{1}{3}$
That means that
$$f(-\frac{1}{3})=\frac{(\frac{8}{9})}{(\frac{10}{9})+\frac{2}{3} \cos \omega}=\frac{4}{5+3 \cos \omega}$$
If we are instead given
$$\frac{4}{5+3 \cos \omega}=\frac{(1-\alpha^2)}{(1+\alpha^2)-2 \alpha \cos \omega},$$
is there a way to determine $\alpha$ by observation? Or, is there a way to find $\alpha$ without simply solving for $\alpha$ and using the quadratic formula?
An example in my textbook does it in a single step; just wondering If I'm missing something.