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Is there an easy way to prove this result?

$$\int\ \cos^2\Big(\arctan\big(\sin\left( \text{arccot}(x)\right)\big)\Big)\ \text{d}x = x - \frac{1}{\sqrt{2}}\arctan\left(\frac{x}{\sqrt{2}}\right)$$

I tried some substitutions but I got nothing helpful, like:

  • $x = \cot (z)$

I also tried the crazy one:

  • $x = \cot(\arcsin(\tan(\arccos(z))))$

Any hint?

Thank you!

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    $\begingroup$ First hint : $\sin(arccot(x))=\frac{1}{\sqrt{x^2+1}}$ (for positive $x$). $\endgroup$
    – Peter
    Sep 14, 2016 at 20:38
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    $\begingroup$ Drawing triangles really helps. $\endgroup$ Sep 14, 2016 at 20:40
  • $\begingroup$ This really looks like the consequence of a series of trigonometric identities, more than some particular integration technique. $\endgroup$
    – b00n heT
    Sep 14, 2016 at 20:40
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    $\begingroup$ Second hint : $\cos(\arctan(x))=\frac{1}{\sqrt{x^2+1}}$ $\endgroup$
    – Peter
    Sep 14, 2016 at 20:43
  • $\begingroup$ @Peter : Could you type $\cos(\arctan(x))$ instead of $\cos(arctan(x))$. The latter doesn't have proper spacing in things like $a\arctan b$ or $a\arctan(b)$ (and I include both of those examples so you can see the context-dependent nature of the spacing. $\qquad$ $\endgroup$ Sep 14, 2016 at 20:44

5 Answers 5

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$\alpha = \cot^{-1} x\\ \beta = \tan^{-1} (\sin \alpha)$

Use trig indentities to find $\csc \alpha$ and $\sin \alpha$

$\sin\alpha = \tan \beta$

Use similar identities to find $\sec \beta$ and $\cos\beta$

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As mentioned in comments, draw the triangles. You have $$ \frac x 1 = x = \cot\theta = \frac{\text{adjacent}}{\text{opposite}} $$ so if you have a triangle in which $\text{opposite}=1$ and $\text{adjacent} = x$ then you have $\text{hypotenuse} = \sqrt{x^2+1}$ and so $$ \sin\theta = \frac{\text{opposite}}{\text{hypotenuse}} = \frac 1 {\sqrt{x^2+1}} $$ so $$ \sin\operatorname{arccot} x = \frac 1 {\sqrt{x^2+1}}. $$ Then do a similar thing with the cosine of the arctangent.

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HINT: $$\cos \left(\arctan \left(x\right)\right)=\frac{\sqrt{1+x^2}}{1+x^2}$$ So $$\begin{align} \\& \int \:\left(\cos \left(\arctan \left(\sin \left(\operatorname{arccot}\left(x\right)\right)\right)\right)\right)^2dx \\& = \int \frac{x^2+1}{x^2+2}dx \\& = \color{red}{\frac{\sqrt{2}x-\arctan \left(\frac{\sqrt{2}x}{2}\right)}{\sqrt{2}}+C} \end{align}$$

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After using trigonometry, you should be able to get

$$\cos(\arctan(\sin( \text{arccot}(x)))) = \sqrt{1 - \frac{1}{x^2+2}}$$

From there, I'd assume it's just a trig sub problem.

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  • $\begingroup$ From there is really trivial. I am now curious to get that identity. Thanks! $\endgroup$
    – Enrico M.
    Sep 14, 2016 at 20:43
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    $\begingroup$ Well, I'll do half of it, the rest of it is done in a very similiar way. Let $\theta = \text{arccot}(x)$, then $\cot(\theta) = x$, and hence using right triangles we've $\sin(\theta) = \sqrt{1+x^2}$. $\endgroup$
    – DaveNine
    Sep 14, 2016 at 20:45
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The integral is equal to $$\int 1-\frac{1}{2+x^2} \, dx$$ this is now easy to integrate.

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  • $\begingroup$ This is a hint. Please mark it as a hint! $\endgroup$
    – Peter
    Sep 14, 2016 at 20:40
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    $\begingroup$ That is cool, but can you prove this identity? Does it holds for all $x$? $\endgroup$
    – Enrico M.
    Sep 14, 2016 at 20:40
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    $\begingroup$ Its not a hint, its a concise solution. $\endgroup$ Sep 14, 2016 at 20:42
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    $\begingroup$ @FourierTransform Yes, for example $\cos^2x=\frac{1}{1+\tan^2x}$, this removes the first iteration, the second is similar. $\endgroup$ Sep 14, 2016 at 20:43

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