Formula for summing an arbitrary number $n$ of cos functions?

Question

$$\cos A+\cos B+\cos C=1+4\sin \frac {A}{2}\sin \frac {B}{2}\sin \frac {C}{2}$$

Is it possible to derive a generalised formula for

$$\cos A+\cos B+\cos C+...+\cos N$$

i.e., a formula for summing an arbitrary number $n$ of $\cos$ functions? Perhaps something along the lines of

$$\cos A+\cos B+\cos C+...+\cos N=1+(n+1)\sin \frac {A}{2}\sin \frac {B}{2}\sin \frac {C}{2}...\sin \frac {N}{2}$$

But that's just a wild guess, and I'd have no idea how to prove it.

You fail to mention that that formula is predicated on $A+B+C=\pi$. — Angina Seng, Apr 04 '18 at 06:42
Is your question also conditioned to $A+B+ \cdots + N = \pi$ ? — Andreas, Apr 04 '18 at 06:43
OK, to clarify, I'm looking for a formula for $\cos A \pi x+...+\cos N \pi x$ — Richard Burke-Ward, Apr 04 '18 at 07:34

score 0 · Answer 1 · answered Apr 04 '18 at 06:54

0

It doesn't seem true in general, let try with some numerical value to disprove that.

For this special case we can refer to Lagrange's trigonometric identities

$$\sum_{n=1}^N \cos (n\theta) = -\frac{1}{2}+\frac{\sin\left(\left(N+\frac{1}{2}\right)\theta\right)}{2\sin\left(\frac{\theta}{2}\right)}$$

answered Apr 04 '18 at 06:54

user

I can see that this disproves my suggested general solution - but I can't see how it helps me arrive at a correct general solution for $\cos A\pi x+\cos B\pi x+...+\cos N\pi x$, because the Lagrange identity only works for $n={1,2,3...N}$, not for arbitrary values of ${A,B,C...N}$. – Richard Burke-Ward Apr 04 '18 at 12:02

1 Answers1