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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 ( 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? – J. M. Sep 4 '12 at 1:33
seems valid....... so are you wanting me to apply the right hand sides and try to show that its z? – Squirtle Sep 4 '12 at 1:39
I presume you know the usual procedure for inverting a function? – J. M. Sep 4 '12 at 1:41
the inverse of a trig function isn't its reciporal .... so I'm not sure I know what you are suggesting. – Squirtle Sep 4 '12 at 1:53
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. – J. M. Sep 4 '12 at 2:02

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