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.


Something like:

$$ \cos (\alpha - \beta)= \cos \alpha \cos \beta + \sin \alpha \sin \beta $$


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. $

  • 1
    $\begingroup$ Are you able to express $\cos\frac{\alpha}{\pi}$ in terms of $\cos(\alpha)$? $\endgroup$ – Jack D'Aurizio May 13 '15 at 20:18
  • $\begingroup$ You can take the power series for $\cos x$ and just plug in $n\alpha$ for $x$. $\endgroup$ – Gregory Grant May 13 '15 at 20:23
  • $\begingroup$ No, it is possible for integral $n$. I mention it only as it occurs to anyone at the beginning during approach..shall edit it. $\endgroup$ – Narasimham May 13 '15 at 20:28
  • $\begingroup$ @Narasimham I think you might want to rewrite the question, because right now it makes no sense. $\endgroup$ – Braindead May 13 '15 at 20:32
  • $\begingroup$ I haven't looked it up but if that's a correct use of the word "tally" I've never heard of it. $\endgroup$ – 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.

  • $\begingroup$ 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 ? $ $\endgroup$ – Narasimham May 13 '15 at 22:07
  • $\begingroup$ I answered for the quotient. The case $\alpha=\beta=x$ is uninformative. $\endgroup$ – 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 $e^\beta$.

  • $\begingroup$ 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. $ $\endgroup$ – Narasimham May 13 '15 at 21:22

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