Problem 49 - Calculus Early Transcendentals: Pearson New International Edition

Question

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 verified as an identity.

Step 1: Define Hyperbolic Functions

Recall that the hyperbolic tangent function is defined as \( \tanh x = \frac{\sinh x}{\cosh x} \). The hyperbolic sine function is \( \sinh x = \frac{e^x - e^{-x}}{2} \) and the hyperbolic cosine function is \( \cosh x = \frac{e^x + e^{-x}}{2} \). We will use these definitions in our verification.

Step 2: Express \( \tanh(x+y) \) using hyperbolic sine and cosine

The sum of angles formula for hyperbolic tangent is \( \tanh(x+y) = \frac{\sinh(x+y)}{\cosh(x+y)} \). Using the addition formulas for hyperbolic sine and cosine, we express \( \sinh(x+y) = \sinh x \cosh y + \cosh x \sinh y \) and \( \cosh(x+y) = \cosh x \cosh y + \sinh x \sinh y \).

Step 3: Substitute Hyperbolic Identities

Substitute the identities into \( \tanh(x+y) \): \[\tanh(x+y) = \frac{\sinh x \cosh y + \cosh x \sinh y}{\cosh x \cosh y + \sinh x \sinh y}\].

Step 4: Express \( \frac{\tanh x + \tanh y}{1 + \tanh x \tanh y} \)

Substitute the definition of \( \tanh x \) and \( \tanh y \): \[\frac{\tanh x + \tanh y}{1 + \tanh x \tanh y} = \frac{\frac{\sinh x}{\cosh x} + \frac{\sinh y}{\cosh y}}{1 + \frac{\sinh x}{\cosh x} \cdot \frac{\sinh y}{\cosh y}}\].

Step 5: Simplify the Right-Hand Side

Combine the fractions: \[\frac{\frac{\sinh x \cosh y + \sinh y \cosh x}{\cosh x \cosh y}}{\frac{\cosh x \cosh y + \sinh x \sinh y}{\cosh x \cosh y}}\]. Simplifying this gives \( \frac{\sinh x \cosh y + \cosh x \sinh y}{\cosh x \cosh y + \sinh x \sinh y} \), which matches \( \tanh(x+y) \).

Step 6: Conclude the Identity

After simplification, both sides of the given equation match. Therefore, it proves that \( \tanh(x+y) = \frac{\tanh x + \tanh y}{1 + \tanh x \tanh y} \) is indeed an identity.

Key Concepts

Hyperbolic Functions

In mathematics, hyperbolic functions are analogs to the ordinary trigonometric functions but for a hyperbola, rather than a circle. Just like sine and cosine are central to circular trigonometry, the hyperbolic sine (\( \sinh x = \frac{e^x - e^{-x}}{2} \)) and hyperbolic cosine (\( \cosh x = \frac{e^x + e^{-x}}{2} \)) form the foundation of hyperbolic trigonometry. These functions are useful in various areas including calculus, special relativity, and complex analysis. Hyperbolic functions share properties analogous to trigonometric ones, for instance, \( \cosh^2 x - \sinh^2 x = 1 \), which mirrors the Pythagorean identity of circular functions.

• They model the shape of a hanging cable (catenary).
• Appear in solutions of the Laplace and heat equations.
• Are used in computations involving complex numbers.

Understanding how these functions relate helps in simplifying expressions and verifying identities like the one in our exercise.

Tanh Function

The hyperbolic tangent function, denoted as \( \tanh x \), is defined as the ratio of the hyperbolic sine and hyperbolic cosine functions: \( \tanh x = \frac{\sinh x}{\cosh x} \). This function behaves similarly to the tangent function in trigonometry. It maps real numbers to the interval \((-1, 1)\) and is an odd function, meaning \( \tanh(-x) = -\tanh(x) \).
Here are some of its important properties:

• As \( x \to \infty \), \( \tanh x \to 1 \)
• As \( x \to -\infty \), \( \tanh x \to -1 \)
• Derivative: \( \frac{d}{dx}(\tanh x) = 1 - (\tanh x)^2 \).

The hyperbolic tangent is particularly useful in solving differential equations and in the field of engineering for systems involving damping or electrical circuits.

Sum of Angles Formula

The sum of angles formula is critical in both circular and hyperbolic trigonometry. For hyperbolic functions, consider the hyperbolic tangent addition formula: \[ \tanh(x+y) = \frac{\tanh x + \tanh y}{1 + \tanh x \tanh y} \]. This formula allows you to express the hyperbolic tangent of a sum of two angles in terms of the hyperbolic tangents of the individual angles.

This identity is verified by:

• Expressing \( \tanh(x+y) \) as \( \frac{\sinh(x+y)}{\cosh(x+y)} \)
• Using the addition formulas for hyperbolic sine \( \sinh(x+y) = \sinh x \cosh y + \cosh x \sinh y \)
• Using the addition formulas for hyperbolic cosine \( \cosh(x+y) = \cosh x \cosh y + \sinh x \sinh y \)

These elegant formulas simplify the work of translating properties across sums of angles and proving identities by transforming them into more manageable forms using hyperbolic identities.