I have a problem, that I am trying to simplify, but there does not seem to be something obvious regarding it.
Very simply, I am trying to figure out if there is a way to 'open' the following:
$$ \tan^{-1} (\frac{a}{b}) - \tan^{-1} (\frac{c}{d}) $$
By 'open', I simply mean that I would like to be able to group $a$, $b$, $c$, and $d$ together somehow, so that I can continue with my work. I have scoured wiki for something that might allow me to do that, but could not find anything.
Disclaimer: I should also add that the $\tan^{-1}$ here refers to the 'Four Quadrant' inverse tangent, usually found in computer programs as 'atan2' instead of simply 'atan'.
Thanks for any insight.
note that: $$\tan(\alpha-\beta )=\frac{\tan\alpha- \tan\beta }{1+\tan\alpha \tan\beta } $$ $$\Longrightarrow \alpha -\beta =\tan^{-1}\left(\frac{\tan\alpha- \tan\beta }{1+\tan\alpha \tan\beta }\right).$$ Try letting $\alpha=\tan^{-1}\left(\frac{a}{b}\right)$ and $\beta =\tan^{-1}\left(\frac{c}{d}\right)$. Then $\tan\alpha=\frac{a}{b}$ and $\tan\beta=\frac{c}{d}$.