The supremum of the function $f(x)=\frac{1-\cos Nx}{1-\cos x}$

Question

I have the following function:

$f(x)=\dfrac{1-\cos Nx}{1-\cos x}$

Where N is integer.

I know the function has Sup when x goes to $2n\pi$ $n\in\mathbb{N}$. But is it possible to show this? Thank you in advanced.

Answer 1

We have:

$$ f(x) = \left(\frac{\sin(Nx/2)}{\sin(x/2)}\right)^2 $$ so $x=0$ is a stationary point, a local maximum, for which $f(x)=N^2$.

Such a maximum is indeed a global maximum, since the only solutions of $$ \frac{\sin^2(Nz)}{\sin^2(z)}=N^2 $$ occur at $z\in\pi\mathbb{Z}$.