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

Question

This post gives

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

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.

Answer 1

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

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