TITLE: Is there a name for sec(x)'s relationship with tan(x)?
QUESTION [7 upvotes]: In a couple of trig identities, esp to do with integrals and derivatives, you see a relationship between tan(x) and sec(x). Similarly between csc(x) and cot(x).
$ \frac{d}{dx}\tan(x) = \sec^2(x) $
$ \frac{d}{dx}\sec(x) = \sec(x) \tan(x) $
$ tan^2(x) + 1 = sec^2(x) $
Is there a name for this apparent relationship between $\tan(x)$ and $\sec(x)$?  Something like "complimentary", or "counterparts" of one another..?

REPLY [11 votes]: I wouldn't say it's "nothing special".  In fact it's quite special.  If $f(x)$ and $g(x)$ are functions that satisfy $f'(x) = g(x)^2$ and $f(x)^2 + 1 = g(x)^2$, then $f'(x) = f(x)^2 + 1$.  But that is a differential equation whose general solution is $f(x) = \tan(x + C)$, where $C$ is an arbitrary constant.  And then we get $g(x)^2 = \tan^2(x+C) + 1 = \sec^2(x+C)$, so $g(x) = \pm \sec(x+C)$.