Difference of Arccotangents

Theorem

$\arccot a - \arccot b = \arccot \dfrac {a b + 1} {a - b}$

where $\arccot$ denotes the arccotangent.

Let $x = \arccot a$ and $y = \arccot b$.

Then:

$\text {(1)}: \quad$

$\ds \cot x$

$=$

$\ds a$

$\text {(2)}: \quad$

$\ds \cot y$

$=$

$\ds b$

$\ds \map \cot {\arccot a - \arccot b}$

$=$

$\ds \map \cot {x - y}$

$\ds $

$=$

$\ds \frac {\cot x \cot y + 1} {\cot x - \cot y}$

$\ds $

$=$

$\ds \frac {a - b} {1 + a b}$

by $(1)$ and $(2)$

$\ds \leadsto \ \ $

$\ds \arccot a - \arccot b$

$=$

$\ds \arccot \frac {a b + 1} {a - b}$

$\blacksquare$