The identity for hyperbolic functions is given by

Hyperbolic functions serve as an analogy to trigonometric functions and are defined using the exponential function $e^x$ and its inverse, $\ln(x)$. The two primary hyperbolic functions are the hyperbolic sine, denoted as $\sinh(x)$, and the hyperbolic cosine, denoted as $\cosh(x)$.

The identity

is reminiscent of the Pythagorean identity for trigonometric functions, which states

To prove the hyperbolic identity, we start with the definitions of $\cosh(x)$ and $\sinh(x)$:

Substituting these definitions into the left-hand side of our identity yields:

Next, we expand the squares and simplify:

Thus, we conclude that the identity

holds true for all values of $x$. This identity is valuable for simplifying expressions that involve hyperbolic functions and has practical applications in fields such as physics and engineering.