So we can suppose that we can find the finite sum
$$\sum_{x=a}^b{ \tan{(x)} + \cot{(x)} } \tag{1}$$
for essentially all integer values of $x$.
I'm wondering, can we derive either:
$$\sum_{x=a}^b{ \tan{(x)} } \qquad\text{ or }\qquad \sum_{x=a}^b{ \cot{(x)} } \tag{2}$$
from this sum?
It may be possible to use substitution for $x$ in (1), but there is no guarantee that we can find the sums for non-integer values of $x$ in (1).
I know that it is possible to find the sums in (2). However, I'm primarily interested in if we can derive either sum in (2) from (1).