Proving the Hyperbolic Function Identity (1+tanhx)/(1-tanhx)=e^(2x)

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Homework Help Overview

The problem involves proving the hyperbolic function identity (1+tanhx)/(1-tanhx)=e^(2x), which falls under the subject area of hyperbolic functions and identities.

Discussion Character

  • Exploratory, Mathematical reasoning

Approaches and Questions Raised

  • The original poster attempts to substitute tanhx with its definitions in terms of exponential functions but expresses uncertainty about how to proceed. Some participants offer hints to simplify the expression further, suggesting rewriting it in a different form.

Discussion Status

The discussion includes attempts to simplify the expression and explore different forms of the hyperbolic functions. While some guidance has been provided, there is no explicit consensus on the approach, and participants are still working through the problem.

Contextual Notes

Participants express difficulty in rewriting the expression to a more recognizable form, indicating potential gaps in understanding or missing connections in hyperbolic function identities.

Pietair
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Homework Statement


Prove that:
(1+tanhx)/(1-tanhx)=e^(2x)

Homework Equations



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The Attempt at a Solution



I tried substituting tanhx for (e^x-e^(-x))/(e^x+e^(-x)) and for (e^(2x)-1)/(e^(2x)+1))

But I really have no clue how to continue...
 
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Hi Pietair! :smile:

Hint : (1+tanhx)/(1-tanhx) = (1 + sinhx/coshx)/(1 - sinhx/coshx) = … ? :wink:
 
Thanks for your answer but it still doesn't make sense.

I don't know how to rewrite it to something more "common".
 
Pietair said:
Thanks for your answer but it still doesn't make sense.

I don't know how to rewrite it to something more "common".

try simplifying (1 + sinhx/coshx)/(1 - sinhx/coshx) …

get rid of the internal fractions :wink:
 
(coshx+sinhx)/(coshx-sinhx)

= (0.5e^x+0.5e^(-x)+0.5e^x-0.5e^(-x))/((0.5e^x+0.5e^(-x)-0.5e^x+0.5e^(-x))

= e^x/e^(-x)

= e^2x (proven)

Thanks a lot!
 

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