Problem 44 - Calculus Early Transcendentals: Pearson New International Edition

Question

Verify that the given equations are identities. \(e^{-2 x}=\cosh 2 x-\sinh 2 x\)

Step-by-Step Solution

Answer

The equation is an identity since both sides simplify to \(e^{-2x}\).

Step 1: Recall Definitions of Hyperbolic Functions

Understand the definitions of hyperbolic functions: The hyperbolic cosine function is given by \[\cosh 2x = \frac{e^{2x} + e^{-2x}}{2}\]and the hyperbolic sine function is given by \[\sinh 2x = \frac{e^{2x} - e^{-2x}}{2}.\]

Step 2: Substitute Definitions into the Right Side

Substitute the hyperbolic functions into the right side of the equation:\[\cosh 2x - \sinh 2x = \left(\frac{e^{2x} + e^{-2x}}{2}\right) - \left(\frac{e^{2x} - e^{-2x}}{2}\right).\]

Step 3: Simplify the Expression

Simplify the expression by combining the terms inside the parentheses:\[\cosh 2x - \sinh 2x = \frac{e^{2x} + e^{-2x}}{2} - \frac{e^{2x} - e^{-2x}}{2} = \frac{e^{2x} + e^{-2x} - e^{2x} + e^{-2x}}{2}.\]

Step 4: Further Simplify to Obtain the Left Side

Combine like terms:\[\frac{e^{2x} - e^{2x} + e^{-2x} + e^{-2x}}{2} = \frac{2e^{-2x}}{2} = e^{-2x}.\]

Step 5: Verify the Identity

Since the right side simplifies to \(e^{-2x}\), which matches the left side of the original equation, the identity is verified.

Key Concepts

Hyperbolic Identities

Hyperbolic identities are mathematical expressions that relate hyperbolic functions in a similar manner to trigonometric identities. They provide a framework for simplifying and transforming expressions involving hyperbolic functions such as hyperbolic sine (\(\sinh\)) and hyperbolic cosine (\(\cosh\)). Exploring these identities helps us understand the properties and relationships between different hyperbolic functions. Some common hyperbolic identities include:

• The identity for hyperbolic cosine and sine: \(\cosh^2(x) - \sinh^2(x) = 1\).
• The double angle identity for hyperbolic sine: \(\sinh(2x) = 2 \sinh(x) \cosh(x)\).
• The double angle identity for hyperbolic cosine: \(\cosh(2x) = \cosh^2(x) + \sinh^2(x)\).

These identities are particularly useful in solving hyperbolic function equations and simplifying expressions, much like how trigonometric identities are used in calculus and geometry. They stem from the definitions of hyperbolic sine and cosine, which we'll explore next.

Cosh and Sinh Definitions

Understanding the definitions of the hyperbolic functions \(\cosh\) (hyperbolic cosine) and \(\sinh\) (hyperbolic sine) is essential to working with hyperbolic identities. The hyperbolic cosine is defined as:\[\cosh(x) = \frac{e^x + e^{-x}}{2}\]This definition mirrors the average of exponential growth and decay, providing a way to model the symmetric nature of \(\cosh\). Similarly, the hyperbolic sine function is defined by:\[\sinh(x) = \frac{e^x - e^{-x}}{2}\]This equation represents the difference between exponential growth and decay, capturing the asymmetric nature of \(\sinh\). Both \(\cosh\) and \(\sinh\) can be thought of as the hyperbolic analogs to cosine and sine in trigonometry, but without periodicity. These definitions are crucial for calculating and verifying hyperbolic identities, as seen in the example where \(e^{-2x}\) is verified using the definitions of \(\cosh 2x\) and \(\sinh 2x\).

Exponential Functions

Exponential functions are critical to understanding hyperbolic functions since they are directly linked through their definitions. An exponential function typically takes the form:\[y = a \, e^{bx} \]where \(e\) is the base of the natural logarithms, approximately equal to 2.71828. These functions are fundamental in modeling situations of growth and decay, such as population growth, radioactive decay, and compound interest.
In hyperbolic functions, exponential expressions combine to predict behaviors similar to those in physics and engineering contexts. For instance:

• Hyperbolic cosine expresses balanced growth: \(\cosh(x) = \frac{e^x + e^{-x}}{2}\).
• Hyperbolic sine depicts differential growth: \(\sinh(x) = \frac{e^x - e^{-x}}{2}\).

Understanding these relationships enables deeper comprehension of not just mathematical theory but also real-world phenomena, enhancing the application range of exponential and hyperbolic functions in problem-solving scenarios.