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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..?

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I'm amused to upvote two answers that explicitly disagree. But I think they both (and particularly in conjunction) help with understanding. Some of mathematics is what we choose to name. – Ross Millikan Apr 6 '11 at 5:18
up vote 9 down vote accepted

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 $g(x) = \pm \sec(x+C)$.

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No, there's no name for this relationship - it's a coincidence. Nothing special is going on in the identities you mentioned. In fact, the existence of a separate name "$\sec(x)$" for the function $\frac{1}{\cos(x)}$ is just a historical accident - the identities you stated are really relationships between $\sin$ and $\cos$, or even more properly, really relationships of the function $e^{ix}$ with itself.

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