To say that ${principal}/{interest} = f(r, term)$ is a bit oversimplified.
If $term$ is counted from the time when the loan originated,
you will need at least one more piece of information, such as the original length of the loan.
I think it is simpler to write
$$
\frac{\Delta principal}{interest} = g(r, k)
$$
where $k$ is the number of remaining payments, including the one for which you want to find the ratio ${\Delta principal}/{interest}$, and $r$ is the rate of interest per payment period.
(And I really would rather write ${\Delta principal}/{interest}$
instead of ${principal}/{interest}$, since "$principal$" seems like it ought to be the amount of principal still owed, in which case ${principal}/{interest}$ is just $1/r$.)
The principal still owed on the loan just prior to the final $k$ payments is the present value of a stream of $k$ equal payments of the amount $Pmt$, which is
$$
PV = \frac{1 - (1 + r)^{-k}}{r} Pmt.
$$
Therefore
$$
\frac{Pmt}{PV} = \frac{r}{1 - (1 + r)^{-k}}.
$$
But
$$
\frac{Pmt}{PV} = \frac{interest + \Delta principal}{PV}
= r + \frac{\Delta principal}{PV}
$$
and therefore
$$
\frac{\Delta principal}{PV} = \frac{Pmt}{PV} - r
= \frac{r}{1 - (1 + r)^{-k}} - r = \frac{r}{(1 + r)^k - 1}.
$$
Finally,
$$
\frac{\Delta principal}{interest}
= \frac{\left(\dfrac{\Delta principal}{PV}\right)}
{\left(\dfrac{interest}{PV}\right)}
= \frac{\left(\dfrac{r}{(1 + r)^k - 1}\right)}{r}
= \frac{1}{(1 + r)^k - 1}.
$$
I notice, however, that in the video they were actually talking about
$$
\frac{\Delta principal}{Pmt}.
$$
This can be derived from
$$
\frac{\Delta principal}{Pmt}
= \frac{\left(\dfrac{\Delta principal}{PV}\right)}
{\left(\dfrac{Pmt}{PV}\right)}
= \frac{\left(\dfrac{r}{(1 + r)^k - 1}\right)}
{\left(\dfrac{r}{1 - (1 + r)^{-k}}\right)}
= \frac{1}{(1 + r)^k}.
$$
