What is the identity for hyperbolic functions?

The identity for hyperbolic functions is cosh^2(x) - sinh^2(x) = 1.

Hyperbolic functions are a set of functions that are analogous to trigonometric functions. They are defined in terms of the exponential function, e^x, and its inverse, ln(x). The two most commonly used hyperbolic functions are the hyperbolic sine (sinh) and hyperbolic cosine (cosh).

The identity for hyperbolic functions is cosh^2(x) - sinh^2(x) = 1. This identity is similar to the Pythagorean identity for trigonometric functions, sin^2(x) + cos^2(x) = 1. To prove this identity, we start with the definitions of cosh and sinh:

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

We can then substitute these definitions into the left-hand side of the identity:

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

Expanding the squares and simplifying, we get:

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

Therefore, the identity cosh^2(x) - sinh^2(x) = 1 holds for all values of x. This identity is useful in simplifying expressions involving hyperbolic functions, and it also has applications in physics and engineering.