Generalized distance formula for projections of arbitrary rank

Let $\mathbb{M}_n(\mathbb{C})$ be the algebra of $n\times n$ complex matrices, let $\mathcal{N}(\mathbb{C}^n)$ be the set of nilpotent operators, and let $\nu_{r,n}$ denote the distance from the set of rank-$r$ projections in $\mathbb{M}_n(\mathbb{C})$ to $\mathcal{N}(\mathbb{C}^n)$. Generalized distance formula. For every $n\in\mathbb{N}$ and each $r\in\{1,2,\ldots,n\}$,

$$\nu_{r,n}=\frac{1}{2}\sec\left(\frac{\pi}{\frac{n}{r}+2}\right).$$

This extends the known distance formulas for projections of rank $1$ and $n-1$, and is motivated by computations for intermediate ranks in dimensions $4$ and $5$; no resolution status is supplied in the source.