What is the double angle formula in hyperbolic functions?

The double angle formula in hyperbolic functions is:

cosh(2x) = cosh^2(x) + sinh^2(x)

This formula relates the hyperbolic cosine of twice an angle to the hyperbolic cosine and hyperbolic sine of the angle.

To derive this formula, we start with the identity:

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

We can rewrite this as:

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

Now, let's consider the hyperbolic cosine of twice an angle:

cosh(2x) = cosh(x + x)

Using the identity for the sum of hyperbolic functions, we get:

cosh(2x) = cosh(x) cosh(x) + sinh(x) sinh(x)

Substituting cosh^2(x) for 1 + sinh^2(x), we get:

cosh(2x) = cosh^2(x) + sinh^2(x)

This is the double angle formula for hyperbolic functions.

This formula can be useful in simplifying expressions involving hyperbolic functions, or in solving hyperbolic equations. For example, if we have an equation involving cosh(2x), we can use the double angle formula to rewrite it in terms of cosh(x) and sinh(x), which may be easier to solve.