Difference of Arctangents/Mistake

Source Work

 * $\text V$. Trigonometry: Formulae $(40)$
 * $\text V$. Trigonometry: Formulae $(40)$

Mistake

 * $\tan^{-1} a - \tan^{-1} b = \tan^{-1} \dfrac {a - b} {1 + a b}$

Correction
This is valid only if $-\dfrac \pi 2 < \tan^{-1} a - \tan^{-1} b < \dfrac \pi 2$.

Outside that range, either:
 * $\size {\tan^{-1} a - \tan^{-1} b} = \dfrac \pi 2$ in which case $\tan^{-1} \dfrac {a + b} {1 - a b}$ is undefined

or:
 * $\size {\tan^{-1} a - \tan^{-1} b} > \dfrac \pi 2$ in which case $\tan^{-1} \dfrac {a - b} {1 + a b}$ is either $\tan^{-1} a - \tan^{-1} b + \pi$ or $\tan^{-1} a - \tan^{-1} b - \pi$ according to whether positive or negative.