Is this expression strictly positive? $n \left(\cos\frac{k\pi}{n}\right)\left(1-\cos\frac{k\pi}{n}\right)-\sin\frac{k\pi}{n}$

Question

Let us define a function $f(k,n)$ by

\begin{equation} f(k,n)=n \left (\cos\frac{k\pi}{n}\right) \left(1-\cos\frac{k\pi}{n}\right) - \sin \frac{k\pi}{n} \end{equation}

where $\frac{k}{n}$ is irreducible with $k,n \in \mathbb{N}$, with $k \leq \left \lfloor{n/2}\right \rfloor$, and $k \geq 2,$ $n \geq 5$.

I suspect that $f(k,n) >0$.

Plugging in a few values of $k$ and $n$ and computing $f(k,n)$ numerically indeed shows that, but how do I prove / disprove this?