How to derive the identity for hyperbolic cosine?

The identity for hyperbolic cosine is derived using the exponential function and its properties.

The hyperbolic cosine function is defined as cosh(x) = (e^x + e^-x)/2. To derive its identity, we start with the definition of cosh(x) and its derivative:

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

We can use the properties of the exponential function to simplify these expressions. First, we note that e^x * e^-x = 1, so we can multiply the numerator and denominator of cosh(x) by e^x to get:

cosh(x) = (e^2x + 1)/(2e^x)

Similarly, we can multiply the numerator and denominator of sinh(x) by e^x to get:

sinh(x) = (e^2x - 1)/(2e^x)

Now we can use these expressions to derive the identity for cosh^2(x) - sinh^2(x). We start with:

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

Simplifying this expression gives:

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

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

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

Therefore, we have derived the identity cosh^2(x) - sinh^2(x) = 1, which is a fundamental property of hyperbolic functions.