## Noliec 2 years ago Does anyone know where you could find the deduction of the inverse of tanh(x) starting from the standard hyperbolic trigonometry definitions? (Or does anyone possibly know how to deduce it from them?)

1. Noliec

2. Noliec

I basically get it, down to tanh(x)=(e^(2x)-1)/(e^(2x)+1)

3. AravindG

It is right isnt it?

4. Noliec

Indeed, but how do you go from here to the inverse of tanh(x)?

5. hartnn

do u know componendo - dividendo ?

6. AravindG

Take tanh x=y. Then apply log on both sides.

7. AravindG

Solve for x.

8. hartnn

if a/b = c/d then (a+b) /(a-b) = (c+d)/ (c-d) did u know this ?

9. Noliec

That's what I did, I didn't get anywhere on the log expression though. Hmm, not sure hartnn.

10. hartnn

that rule is called componendo-dividendo apply that for y/1= (e^(2x)-1)/(e^(2x)+1)

11. Noliec

I'd have to give a proof in order to utilize it though. D: It's supposed to be solved without it.

12. hartnn

that will isolate e^2x and hence you can apply log

13. hartnn

solved without what ?

14. Noliec

No shortcuts. :D

15. Noliec

Algebraically

16. AravindG

Great! Good work!

17. hartnn

lol ok, don't apply that rule (bdw, that was algebraic property) y= (e^(2x)-1)/(e^(2x)+1) y (e^2x +1) = e^2x -1 y e^2x + y = e^2x -1 y e^2x -e^2x = -1-y need i go further ?

18. hartnn

hint for next step factor out e^2x from on left side

19. hartnn

but did u get what i did ?

20. hartnn

$$\huge \dfrac{1}{2}\ln (\dfrac{1+x}{1-x})$$ you should get this

21. hartnn

-1-y / y-1 = (1+y)/ (1-y) the 1/2 will come outside ln right ? because 2x= ln (...)

22. Noliec

What happens after you apply ln?

23. hartnn

e^2x =[(1+y)/(1-y) ] clear till here ??

24. Noliec

Man I feel stupid. D:

25. Noliec

Absolutely.

26. hartnn

any more doubts?

27. Noliec

Nope, thanks a lot! :)

28. hartnn

welcome ^_^