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.
EDIT1:
Something like:
$$ \cos (\alpha - \beta)= \cos \alpha \cos \beta + \sin \alpha \sin \beta $$
EDIT2:
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. $
$\endgroup$ 92 Answers
$\begingroup$$\cos(\alpha)$ and $\cos(\beta)$ are periodic functions.
Now assume that $\cos(\alpha\beta)=f(\cos(\alpha),\cos(\beta))$ and set $\alpha=\beta=x$. Then $\cos(x^2)=f(\cos(x),\cos(x))$ must be a periodic function, which is obviously false. So such a formula cannot exist.
Similar reasoning with $\cos((x+1)/x)$ excludes a formula for division.
This contrast with the case of addition, for which "$\cos(2x)$ must be periodic" raises no contradiction.
$\endgroup$ 2 $\begingroup$$\cos(n\alpha)$ is indeed expressible as a polynomial in terms of $\cos(\alpha)$ (and $\sin(\alpha)$). Reciprocally, you can in some cases solve that polynomial to obtain $\cos(\alpha/n)$ in terms of $\cos(\alpha)$.
But there is nothing like formulas expressing $\cos(\alpha\beta)$ or $\cos(\alpha/\beta)$ in terms of $\cos(\alpha)$ and $\cos(\beta)$.
Just like $e^{\alpha\beta}$ and $e^{\alpha/\beta}$ are not expressible in terms of $e^\alpha$ and $e^\beta$.
$\endgroup$ 1
