Problem 381 - Calculus Volume 1

Question

Derive \(\cosh ^{2}(x)+\sinh ^{2}(x)=\cosh (2 x)\) from the definition.

Step-by-Step Solution

Answer

\(\cosh^2(x) + \sinh^2(x) = \cosh(2x)\) is true based on hyperbolic definitions.

Step 1: Define Hyperbolic Functions

The hyperbolic cosine and sine functions are defined as follows: \(\cosh(x) = \frac{e^x + e^{-x}}{2}\) and \(\sinh(x) = \frac{e^x - e^{-x}}{2}\). We will use these definitions to solve the problem.

Step 2: Compute \(\cosh^2(x)\)

Find \(\cosh^2(x)\) by squaring the definition: \(\cosh^2(x) = \left(\frac{e^x + e^{-x}}{2}\right)^2 = \frac{e^{2x} + 2 + e^{-2x}}{4}\).

Step 3: Compute \(\sinh^2(x)\)

Find \(\sinh^2(x)\) by squaring the definition: \(\sinh^2(x) = \left(\frac{e^x - e^{-x}}{2}\right)^2 = \frac{e^{2x} - 2 + e^{-2x}}{4}\).

Step 4: Add \(\cosh^2(x)\) and \(\sinh^2(x)\)

Combine \(\cosh^2(x)\) and \(\sinh^2(x)\) by adding them: \(\cosh^2(x) + \sinh^2(x) = \frac{e^{2x} + 2 + e^{-2x}}{4} + \frac{e^{2x} - 2 + e^{-2x}}{4}\).

Step 5: Simplify the Result

Simplify the expression from Step 4: \(\cosh^2(x) + \sinh^2(x) = \frac{(e^{2x} + e^{2x}) + (e^{-2x} + e^{-2x})}{4} = \frac{e^{2x} + e^{-2x}}{2}\).

Step 6: Identify the Result

Notice that \(\frac{e^{2x} + e^{-2x}}{2}\) is the definition of \(\cosh(2x)\). Thus, \(\cosh^2(x) + \sinh^2(x) = \cosh(2x)\).

Key Concepts

Understanding Hyperbolic Cosine

The hyperbolic cosine function, denoted as \( \cosh(x) \), is crucial in understanding certain mathematical models involving hyperbolic shapes. It is defined in terms of exponential functions: \( \cosh(x) = \frac{e^x + e^{-x}}{2} \). This function is quite similar to its trigonometric counterpart, the cosine, but it operates in a hyperbolic context.

Here are some key insights about the hyperbolic cosine:

• Unlike the classic cosine function, \( \cosh(x) \) is always positive. This is due to the exponential terms never being negative.
• \( \cosh(x) \) is an even function, meaning \( \cosh(-x) = \cosh(x) \), which reflects its symmetry about the y-axis.
• Hyperbolic cosine plays a role in describing the shape of a hanging cable, known as a catenary, and appears in the solutions to the equations of motion in special relativity.

A deep understanding of \( \cosh(x) \) helps in grasping complex hyperbolic identities and applying them to solve real-world problems.

Exploring Hyperbolic Sine

The hyperbolic sine function, denoted as \( \sinh(x) \), is another major hyperbolic function, playing a crucial role alongside \( \cosh(x) \). It is defined as \( \sinh(x) = \frac{e^x - e^{-x}}{2} \). Though distinct, it mirrors the trigonometric sine function but in a hyperbolic sense.

Key features and properties include:

• The function is odd, which means \( \sinh(-x) = -\sinh(x) \), signifying that it is symmetric with respect to the origin.
• \( \sinh(x) \) can be thought of as describing the rate of exponential growth and decay, thanks to its positive and negative exponential terms.
• This function is widely used in calculus, geometry, and physics, particularly in scenarios involving hyperbolic equations.

Mastering \( \sinh(x) \) introduces students to many significant mathematical concepts like those found in hyperbolic identities and integrals.

Mathematical Derivation of Hyperbolic Identities

Deriving equations and identities in hyperbolic functions requires a strong grasp of basic functions and algebraic manipulations. This derivation process reflects the fundamental interplay between \( \cosh(x) \) and \( \sinh(x) \).

Consider the derivation of the identity \( \cosh^2(x) + \sinh^2(x) = \cosh(2x) \). The step-by-step derivation stems from their definitions:

• Start by squaring both \( \cosh(x) \) and \( \sinh(x) \) based on their initial formulas.
• Add these squared forms to obtain: \[ \cosh^2(x) + \sinh^2(x) = \frac{(e^{2x} + 2 + e^{-2x})}{4} + \frac{(e^{2x} - 2 + e^{-2x})}{4} \]
• This simplifies the result to \( \frac{e^{2x} + e^{-2x}}{2} \), aligning perfectly with the definition of \( \cosh(2x) \).

This identity showcases how hyperbolic functions are interwoven and how such mathematical derivations reveal deeper truths about their nature.