Problem 50 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 41-52, verify that the given equations are identities. \(\tanh (x-y)=\frac{\tanh x-\tanh y}{1-\tanh x \tanh y}\)

Step-by-Step Solution

Answer

The equation is an identity: \( \tanh(x-y) = \frac{\tanh x - \tanh y}{1 - \tanh x \tanh y} \).

Step 1: Recall the definition of hyperbolic tangent

The hyperbolic tangent function, denoted as \( \tanh z \), is defined by the equation \( \tanh z = \frac{\sinh z}{\cosh z} \). Here, \( \sinh z \) is the hyperbolic sine of \( z \) and \( \cosh z \) is the hyperbolic cosine of \( z \). This definition is crucial for understanding the identity we are tasked to verify.

Step 2: Understand the identity to verify

The identity given is \( \tanh(x-y) = \frac{\tanh x - \tanh y}{1 - \tanh x \tanh y} \). This identity states that the hyperbolic tangent of the difference of two values, \( x \) and \( y \), is equal to a specific quotient involving their individual hyperbolic tangents.

Step 3: Apply the difference formula

Utilizing the hyperbolic tangent difference formula, we express \( \tanh(x-y) \) using \( \sinh \) and \( \cosh \): \[ \tanh(x-y) = \frac{\sinh(x-y)}{\cosh(x-y)} \]Apply the difference identities for \( \sinh \) and \( \cosh \):\[ \sinh(x-y) = \sinh x \cosh y - \cosh x \sinh y \]\[ \cosh(x-y) = \cosh x \cosh y - \sinh x \sinh y \]

Step 4: Substitute and simplify

Substituting these into \( \tanh(x-y) = \frac{\sinh(x-y)}{\cosh(x-y)} \), we get:\[\tanh(x-y) = \frac{\sinh x \cosh y - \cosh x \sinh y}{\cosh x \cosh y - \sinh x \sinh y}\]Recognize that \( \tanh x = \frac{\sinh x}{\cosh x} \) and \( \tanh y = \frac{\sinh y}{\cosh y} \). Use these to simplify the numerator and the denominator.

Step 5: Factor and verify equivalence

In the numerator, it simplifies to:\[\sinh x \cosh y - \cosh x \sinh y = \cosh x \cosh y (\tanh x - \tanh y)\]In the denominator, it simplifies to:\[\cosh x \cosh y - \sinh x \sinh y = \cosh x \cosh y (1 - \tanh x \tanh y)\]Thus, substituting these simplified expressions gives:\[\frac{\cosh x \cosh y (\tanh x - \tanh y)}{\cosh x \cosh y (1 - \tanh x \tanh y)} = \frac{\tanh x - \tanh y}{1 - \tanh x \tanh y}\]This confirms the given identity is correct.

Key Concepts

Hyperbolic Functions

Hyperbolic functions are similar to trigonometric functions but are defined using exponential functions. Just like sine and cosine have their hyperbolic counterparts in the form of hyperbolic sine (\( \sinh z \)) and hyperbolic cosine (\( \cosh z \)), these functions model hyperbolas rather than circles. You can think of them as the hyperbolic analogs of the circular trigonometric functions you might be more familiar with.
One of the most common uses of hyperbolic functions is in solving problems related to hyperbolas in geometry. In engineering, they occur naturally in describing processes that exhibit a certain type of symmetry or exponential behavior.

• \( \sinh z = \frac{e^z - e^{-z}}{2} \)
• \( \cosh z = \frac{e^z + e^{-z}}{2} \)
• \( \tanh z = \frac{\sinh z}{\cosh z} \)

These functions are crucial for understanding hyperbolic identities and equations, which often arise in advanced mathematics and physics.

Hyperbolic Tangent

The hyperbolic tangent function, represented as \( \tanh z \), is defined by the ratio of the hyperbolic sine and hyperbolic cosine of \( z \). It is written as:\[ \tanh z = \frac{\sinh z}{\cosh z} \]Similar to the tangent function in trigonometry, the hyperbolic tangent can describe the shape and structure of hyperbolas. In many practical applications, it is used to model growth processes and hyperbolic curves. This is evident in the identity involving the hyperbolic tangent function which shows how complex expressions can be simplified.

Identity Verification Using Hyperbolic Tangent

In this identity, we look at the hyperbolic tangent of a difference, \( \tanh(x-y) \), and express it using individual hyperbolic tangents of \( x \) and \( y \):

\[ \tanh(x-y) = \frac{\tanh x - \tanh y}{1 - \tanh x \tanh y} \]

This identity is quite useful for transforming products of hyperbolic functions into algebraic sums, making complex calculations much easier.

Trigonometric Identities

Trigonometric identities offer powerful tools for simplifying trigonometric expressions and solving equations. Much like their hyperbolic counterparts, these identities help reduce complex expressions to simpler forms. The hyperbolic tangent identity is one such transformation tool, enabling the simplification into recognizable patterns.
Trigonometric identities include Pythagorean identities, angle sum identities, and double angle identities, among others. The basic trigonometric identities, for instance, involve the relationships between sine, cosine, and tangent functions:

• \( \cos^2 \theta + \sin^2 \theta = 1 \)
• \( \tan \theta = \frac{\sin \theta}{\cos \theta} \)

Using identities like these in calculations allows mathematicians and scientists to develop mathematical models that describe phenomena efficiently. The hyperbolic equivalent, which features identities such as \( \cosh^2 z - \sinh^2 z = 1 \), maintains a similar structure, allowing for an analogous approach to solving and simplifying problems involving hyperbolic functions.