Talk:Sum of Arctangents

Range of validity

This identity is not valid for any $a$, $b$. This is obvious as the right-hand side is bound to be in the interval $(-\pi/2,+\pi/2)$, while the left-hand side ranges in $(-\pi,+\pi)$. In the proof, the incorrect step is the last one, as it is not true that $\arctan\tan(\alpha) = \alpha$ if $|\alpha| \geqslant \pi/2$.

I believe this identity is valid if $ab < 1$. It takes a similar (but different) form if $ab > 1$. I can add the missing part, though... I am new and I don't know an "elegant" proof, other than treating separately each case.

The same comment applies to the Difference of Arctangents page.

Daniele Bjørn Malesani (talk) 09:33, 1 March 2023 (UTC)

Good call. Might be able to fix this by using a modulo $\pi$ technique (although constraining the range to be $\openint {-\dfrac \pi 2} {\dfrac \pi 2}$ instead of $\openint 0 \pi$, and deeming it undefined if $ab=1$).

You are more than welcome to have a go at resolving this. It's a bad mistake and really ought to have been caught in the source book cited. The latter needs to have an errata page raised for it. I will be on the case. --prime mover (talk) 10:08, 1 March 2023 (UTC)

Thanks for the quick feedback. I'll make a note to add some content to this page, though it may take a little time :) Daniele Bjørn Malesani (talk) 10:51, 1 March 2023 (UTC)