Equivalence of Definitions of Real Area Hyperbolic Cosine

Theorem
The following definitions of the real inverse hyperbolic cosine are equivalent:

Definition 1 implies Definition 2
Let $x = \cosh \left({y}\right)$, where $y > 0$.

Let $z = e^y$.

Then:

Also, from Minimum of Real Hyperbolic Cosine Function:
 * $x = \cosh \left({y}\right) \ge 1$

Also:

Thus $x - \sqrt{x^2 - 1}$ is (strictly) positive.

$x - \sqrt{x^2 - 1} > 1$.

Then:

and a contradiction is deduced.

Therefore:
 * $x - \sqrt{x^2 - 1} < 1$.

From Logarithm is Strictly Increasing:
 * $y = \ln \left({x - \sqrt{x^2 - 1} }\right) < \ln 1 = 0$

Since $y$ is (strictly) positive from the first definition of real inverse hyperbolic cosine:
 * $y = \ln \left({x + \sqrt{x^2 - 1}}\right)$

Definition 2 implies Definition 1
Let $z = x + \sqrt{x^2 - 1}$.

Then, $y = \ln z$.

If $-1 < x < 1$, $z$ is not defined.

If $x \le -1$:

If $x \ge 1$, $z \ge 1$.

Therefore, $y = \ln z \ge \ln 1 = 0$.

Therefore:

Also see

 * Equivalence of Definitions of Real Inverse Hyperbolic Sine
 * Equivalence of Definitions of Real Inverse Hyperbolic Tangent
 * Equivalence of Definitions of Real Inverse Hyperbolic Cosecant
 * Equivalence of Definitions of Real Inverse Hyperbolic Secant
 * Equivalence of Definitions of Real Inverse Hyperbolic Cotangent