Prove : coth^2 - 1 = csch^2x

Question

Prove : coth² (x) - 1 = csc² (x)

 

cosh²                           1

____    -   1    =   ________  

sinh²                         sinh²

 

 

Then I substituted in the definitions of sinh and cosh. I won't list all the steps because it's a lot of typing.

 

 

2e^x + 2x + e^-2x      -  1    =    4

____________                    _________

2e^x - 2x + e^-2x                    2e^x - 2x + e^-2x

 

 

I tried isolating the x on one side but that made a total mess. Is there a better way to approach this?

 

 

#13 on p 262

 

Answer 1

It is easy if you use the identity:

 

cosh²x - sinh²x = 1

 

Then:

 

coth²x - 1 = cosh²x / sinh²x - 1 = (cosh²x - sinh²x) / sinh²x = 1 / sinh²x = csch²x