Question

Identities Prove each identity using the definitions of the hyperbolic functions. \(\cosh 2 x=\cosh ^{2} x+\sinh ^{2} x(\text {Hint}:\) Begin with the right side of the equation.)

Short answer

Question: Prove that \(\cosh 2 x=\cosh ^{2} x+\sinh ^{2} x\), using the definitions \(\cosh x = \frac{e^x + e^{-x}}{2}\) and \(\sinh x = \frac{e^x - e^{-x}}{2}\). Answer: By calculating the squares of the given hyperbolic functions, adding the squares together, and simplifying the expression, we have proven that \(\cosh 2 x=\cosh ^{2} x+\sinh ^{2} x\).

Step-by-step solution

Find the square of the hyperbolic function

First, find \(\cosh ^{2} x\) and \(\sinh ^{2} x\) using the given definitions: \(\cosh ^{2} x = \left(\frac{e^x + e^{-x}}{2}\right)^2 = \frac{(e^x + e^{-x})^2}{4}\) \(\sinh ^{2} x = \left(\frac{e^x - e^{-x}}{2}\right)^2 = \frac{(e^x - e^{-x})^2}{4}\)

Calculate the sum of the squares

Calculate the sum of these squares: \(\cosh ^{2} x+\sinh ^{2} x = \frac{(e^x + e^{-x})^2}{4} + \frac{(e^x - e^{-x})^2}{4}\)

Simplify the sum

Simplify the sum using the common denominator: \(\cosh ^{2} x+\sinh ^{2} x = \frac{(e^x + e^{-x})^2 + (e^x - e^{-x})^2}{4}\)

Expand the squares and simplify

Expand the squares and simplify the expression: \(\begin{aligned} \cosh ^{2} x+\sinh ^{2} x &= \frac{e^{2 x} + 2 e^x e^{-x} + e^{-2 x} + e^{2 x} - 2 e^x e^{-x} + e^{-2 x}}{4} \\ &= \frac{2 e^{2 x} + 2 e^{-2 x}}{4} \end{aligned}\)

Simplify and rewrite in terms of hyperbolic functions

Simplify the expression and rewrite it in terms of hyperbolic functions: \(\cosh ^{2} x+\sinh ^{2} x = \frac{e^{2 x} + e^{-2 x}}{2} = \cosh 2 x\) Thus, we have proved the identity \(\cosh 2 x=\cosh ^{2} x+\sinh ^{2} x\).

Key concepts

cosh squared

Hyperbolic cosine, denoted as \(\cosh x\), is a function that emerges as an analog of the trigonometric cosine function, but for a hyperbola rather than a circle. The square of hyperbolic cosine, \(\cosh^2 x\), is crucial in understanding hyperbolic identities.

When we express \(\cosh^2 x\) based on exponential functions, we have \(\cosh^2 x = \left(\frac{e^x + e^{-x}}{2}\right)^2\). This relationship forms the foundation for proving many hyperbolic identities, including the one in our exercise. It mirrors the familiar Pythagorean identity from trigonometry but adapted for hyperbolic functions, offering insights into the geometrical interpretation of hyperbolic angles and distances on a hyperbola.

sinh squared

Hyperbolic sine, represented as \(\sinh x\), is another fundamental hyperbolic function akin to the usual trigonometric sine but again, defined for a hyperbola. Analogously, \(\sinh^2 x\) is the square of hyperbolic sine.

The function can be articulated using exponential functions as \(\sinh^2 x = \left(\frac{e^x - e^{-x}}{2}\right)^2\). This expression is pivotal in the derivation of hyperbolic function identities, and when added to \(\cosh^2 x\), reveals interesting relationships like the one we explore in the textbook exercise, allowing students to draw parallels with the Pythagorean identities they might already know from trigonometry.

exponential functions

Exponential functions, such as \(e^x\), where \(e\) is Euler's number, are indispensable in diverse mathematical disciplines. They are characterized by a constant base raised to a variable exponent and exhibit growth or decay characteristics based on the exponent's sign. Exponential functions are used extensively in the definitions of hyperbolic functions.

In our exercise, we utilize the exponentials \(e^x\) and \(e^{-x}\) to express the squared hyperbolic functions \(\cosh^2 x\) and \(\sinh^2 x\). This usage embodies the exponential nature of hyperbolic functions and underscores their relationship to real-life phenomena such as population growth, radioactive decay, and even the distribution of heat over time.

algebraic simplification

Algebraic Power of Manipulation

When solving mathematical problems, the art of algebraic simplification is a cornerstone technique. It involves restructuring and minimizing expressions without altering their value. This process often includes finding common denominators, expanding products, and canceling terms.

As we step through the textbook example, algebraic simplification is a key maneuver that helps us progress from the expanded form of \(\cosh^2 x + \sinh^2 x\) to the more compact and recognizable \(\cosh 2x\). This ability to manipulate algebraic expressions paves the way to proving identities, solving equations, and simplifying complex formulas, demonstrating the powerful utility of algebra in mathematics.