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I would like to derive the following expression for inverse cotangent:

$\cot^{-1} (z) = \frac{i}{2}[\ln(\frac{z-i}{z}) - \ln(\frac{z+i}{z})]$

But I don't want to take it as "definition" as this page (http://mathworld.wolfram.com/InverseCotangent.html) seems to suggest that, that is the 'standard' defintion (or at least a practical one).

In essence, I'm looking for a definition which only relies on tangent, inverse tangent, and cotangent.

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    I don't understand your requirement. In any event, would you allow the use of $$\cot\,z=i\frac{\exp(iz)+\exp(-iz)}{\exp(iz)-\exp(-iz)}$$ as a definition for cotangent?2012-09-04
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    seems valid....... so are you wanting me to apply the right hand sides and try to show that its z?2012-09-04
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    I presume you know the usual procedure for inverting a function?2012-09-04
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    the inverse of a trig function isn't its reciporal .... so I'm not sure I know what you are suggesting.2012-09-04
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    Would you know how to obtain $x=\pm\sqrt{y}$ from $y=x^2$? That's what I meant when I asked you if you know how to invert a function.2012-09-04
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    Well I think that the general approach is G(H(x)) = H(G(x))=x shows that G and H are inverses. I don't know what else to do.... and such an approach is very messy, it seems.2012-09-04
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    I got the answer.... well, I figure I should post it because this is what we should do, but I can't be bothered to write out all the details... is there a way to post an attachment, or can I delete the question?2012-09-06
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    You could delete it, but it would be more helpful to at least write a sketch of what you did, so it can be checked for correctness.2012-09-06

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