# How is $\cos (\alpha / \beta)$ expressed in terms of $\cos \alpha$ and $\cos \beta$?

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 $\cos \beta.$

• Are you able to express $\cos\frac{\alpha}{\pi}$ in terms of $\cos(\alpha)$? – Jack D'Aurizio May 13 '15 at 20:18
• You can take the power series for $\cos x$ and just plug in $n\alpha$ for $x$. – Gregory Grant May 13 '15 at 20:23
• No, it is possible for integral $n$. I mention it only as it occurs to anyone at the beginning during approach..shall edit it. – Narasimham May 13 '15 at 20:28
• @Narasimham I think you might want to rewrite the question, because right now it makes no sense. – Braindead May 13 '15 at 20:32
• I haven't looked it up but if that's a correct use of the word "tally" I've never heard of it. – Matt Samuel May 13 '15 at 20:36

$\cos(\alpha)$ 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.
• Indeed so.. .but I was asking about quotient argument in particular. For x=y, $\cos (x/y) = \cos( 1 radian) = f( \cos(x) )$, many values f can be found for it, Right ? $– Narasimham May 13 '15 at 22:07 • I answered for the quotient. The case$\alpha=\beta=x$is uninformative. – Yves Daoust May 14 '15 at 8:47$\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$. • I know it. just want to know why$ \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. \$ – Narasimham May 13 '15 at 21:22