Difference of Arctangents/Mistake
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Source Work
1953: L. Harwood Clarke: A Note Book in Pure Mathematics:
- $\text V$. 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.
Sources
- 1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text V$. Trigonometry: Formulae $(40)$