Question

Prove the following identity. ?

Answer 1

This follows almost directly from the definition of hyperbolic functions, as shown below

Explanation:

The hyperbolic functions are defined by

`cosh x = {e^x+e^-x}/2, sinh x = {e^x-e^-x}/2`

So

`cosh^2 x-sinh^2 x = (cosh x-sinh x)(cosh x+ sinh x) = e^x timese^-x =1`

Dividing both sides by `sinh^2 x` gives

`coth^2 x - 1 = csch^2 x`