If it is possible to express $ \cos n \alpha $ in terms of $ \cos \alpha $ as a power series for integer $n$ ...
I like to see an expression for the quotient angle that obviously tallies when $ (\alpha , \beta) $ are swapped.
$$ \cos (\alpha - \beta)= \cos \alpha \cos \beta + \sin \alpha \sin \beta $$
Like to know why $ \cos ( \alpha / \beta )$ cannot be expressed in terms of $ \cos \alpha, \cos \beta $, but $ \cos ( \alpha + \beta) $ can be expressed in terms of $\cos \alpha $ and $ \cos \beta. $