How to derive the identity for hyperbolic functions?

The identity for hyperbolic functions is derived using the definitions of sinh, cosh and tanh.

The hyperbolic functions are defined as follows:

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

We can use these definitions to derive the identity:

cosh^2(x) - sinh^2(x) = 1

Starting with the left-hand side:

cosh^2(x) - sinh^2(x) = [(e^x + e^-x)/2]^2 - [(e^x - e^-x)/2]^2
= [(e^2x + 2 + e^-2x)/4] - [(e^2x - 2 + e^-2x)/4]
= (4/e^2x)[(e^2x + 2 + e^-2x)/4 - (e^2x - 2 + e^-2x)/4]
= (4/e^2x)[(4/e^2x)(2)]
= 1

Therefore, cosh^2(x) - sinh^2(x) = 1, which is the identity for hyperbolic functions.