Question

Prove sinh²+cosh²x=cosh2x?

Answer 1

Start with the definitions of `sinh` and `cosh`:

`sinh x = (e^x-e^x)/2` and `cosh x = (e^x+e^x)/2`

Then the LHS becomes:

`LHS = ((e^x-e^x)/2)^2 + ((e^x+e^x)/2)^2`

`\ \ \ \ \ \ \ \ = (e^(2x)-2+e^(-2x))/4 + (e^(2x)+2+e^(-2x))/4`

`\ \ \ \ \ \ \ \ = (e^(2x)-2+e^(-2x)+e^(2x)+2+e^(-2x))/4`

`\ \ \ \ \ \ \ \ = (2e^(2x)+2e^(-2x))/4`

`\ \ \ \ \ \ \ \ = (e^(2x)+e^(-2x))/2`

`\ \ \ \ \ \ \ \ = cosh (2x) \ \` QED