The derivate of $\tan x$ is very simple obtainable by productrule and can be presented in two forms

$$\frac{d}{dx}\tan x=\frac{1}{\cos^2x}=1+\tan^2x$$

However, i've asked myself has any of these two forms any advantage over the other. (Im asking because i write my own differential-function)

Both can be easily converted into another but thats not the question here.

Both have an comparable easy way to differentiate again and do not build simpler/more complex forms

$\frac{d^2}{dx^2}\tan x=2\frac{\tan x}{\cos^2x}=2\tan(x)\cdot(1+\tan^2x)$

So does any of the two do have have an advantage or something which makes it more appropriate for a transformation, a calculation or something like that which would not be that obvious in the other form?

Perhaps this question was answered before by many CAS-products. Mathematica uses $\dfrac{1}{\cos^2x}=\sec^2x$ probably because its the most readable. What does Maple do and any other CAS you have? (I only have access to Mathematica)

  • 2
    $\begingroup$ I don't see how this question can possibly have an answer without posing it in some more specific context. $\endgroup$
    – Brick
    Oct 3, 2016 at 18:55
  • $\begingroup$ Okay let me phrase it that way: what do you like more to work with. Wenn you differentiate $\tan$, which form do you use? $\endgroup$ Oct 3, 2016 at 19:02
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    $\begingroup$ There are some advantages for the $\sec^2$ form so let's provide this example for the $1+\tan^2$ form. $\endgroup$ Oct 3, 2016 at 19:33
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    $\begingroup$ It's somehow a strange question, therefore I will give you (perhaps) a strange answer: If you derivate a function $f(x)$ and the result can be expressed by $f(x)$ then you have a nice and very clear form of a differential equation. Characteristics of the function can be better understood as if other functions are involved. $\endgroup$
    – user90369
    Oct 3, 2016 at 19:37


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