Problem 12 - Calculus Early Transcendentals

Question

Verify each identity using the definitions of the hyperbolic functions. \(\tanh (-x)=-\tanh x\)

Step-by-Step Solution

Answer

Question: Verify the identity \(\tanh(-x) = -\tanh(x)\). Solution: Using the definitions of hyperbolic functions and their properties, we have shown that \(\tanh(-x) = \frac{-\sinh(x)}{\cosh(x)}\) and \(-\tanh(x) = -\frac{\sinh(x)}{\cosh(x)}\). Since these expressions are identical, it verifies the identity of \(\tanh(-x) = -\tanh(x)\).

Step 1: Define the \(\tanh(x)\) function using hyperbolic functions

Recall that the \(\tanh(x)\) function is defined as the ratio of the hyperbolic sine and hyperbolic cosine functions as follows: \(\tanh(x) = \frac{\sinh(x)}{\cosh(x)}\)

Step 2: Find \(\tanh(-x)\) using the definition of \(\tanh(x)\)

Now, let us find \(\tanh(-x)\) using the definition of \(\tanh(x)\): \(\tanh(-x) = \frac{\sinh(-x)}{\cosh(-x)}\)

Step 3: Apply the properties of hyperbolic functions to \(\sinh(-x)\) and \(\cosh(-x)\)

We know that the hyperbolic sine function is odd and the hyperbolic cosine function is even. It means, \(\sinh(-x) = -\sinh(x)\) and \(\cosh(-x) = \cosh(x)\) Now substitute these in the expression for \(\tanh(-x)\): \(\tanh(-x) = \frac{-\sinh(x)}{\cosh(x)}\)

Step 4: Find \(-\tanh(x)\) using the definition of \(\tanh(x)\)

Now, let us find the expression for \(-\tanh(x)\): \(-\tanh(x) = -\frac{\sinh(x)}{\cosh(x)}\)

Step 5: Compare \(\tanh(-x)\) and \(-\tanh(x)\)

From steps 3 and 4, we have the expressions for \(\tanh(-x)\) and \(-\tanh(x)\) as follows: \(\tanh(-x) = \frac{-\sinh(x)}{\cosh(x)}\) and \(-\tanh(x) = -\frac{\sinh(x)}{\cosh(x)}\) As both expressions are identical, we can conclude that the given identity is true: \(\tanh(-x) = -\tanh(x)\)

Key Concepts

Understanding the Tanh Function

The hyperbolic tangent function, denoted as \( \tanh(x) \), is one of the primary hyperbolic functions. Much like its trigonometric counterpart, the tangent function, \( \tanh(x) \) has its own unique characteristics.
It is defined as the ratio of the hyperbolic sine function, \( \sinh(x) \), to the hyperbolic cosine function, \( \cosh(x) \). In mathematical terms, it's expressed as:

• \( \tanh(x) = \frac{\sinh(x)}{\cosh(x)} \)

This definition means \( \tanh(x) \) is inherently linked to both \( \sinh(x) \) and \( \cosh(x) \), sharing their algebraic properties.
An important property of the \( \tanh(x) \) function is that it is odd. This means it satisfies the identity:

• \( \tanh(-x) = -\tanh(x) \)

This reflects a vertical symmetry about the origin, similar to the odd nature of the regular tangent function.

Delving into Hyperbolic Identities

Hyperbolic identities are equations that are analogous to trigonometric identities, but they apply to hyperbolic functions. These identities are very useful in simplifying and transforming expressions involving hyperbolic functions.
One key identity involving \( \tanh(x) \) is the one verified in the exercise, \( \tanh(-x) = -\tanh(x) \). It showcases the odd nature of the hyperbolic tangent function.
Other common hyperbolic identities include:

• \( \sinh^2(x) + 1 = \cosh^2(x) \)
• \( 1 - \tanh^2(x) = \text{sech}^2(x) \)

These identities help to solve a wide array of mathematical problems and simplify complex expressions. By knowing these, one can transition fluidly between different forms of equations involving hyperbolic functions.

Properties of Sinh and Cosh Functions

The hyperbolic sine and cosine functions, \( \sinh(x) \) and \( \cosh(x) \), are foundational to understanding hyperbolic functions like \( \tanh(x) \).

### Sinh Function\( \sinh(x) \) is defined as:

• \( \sinh(x) = \frac{e^x - e^{-x}}{2} \)

It is an odd function, meaning \( \sinh(-x) = -\sinh(x) \). This property can help when verifying identities like \( \tanh(-x) = -\tanh(x) \).

### Cosh Function\( \cosh(x) \) is defined as:

• \( \cosh(x) = \frac{e^x + e^{-x}}{2} \)

It is an even function, satisfying \( \cosh(-x) = \cosh(x) \), which means it exhibits symmetry about the y-axis.
As the ratio of \( \sinh(x) \) to \( \cosh(x) \) creates the \( \tanh(x) \) function, understanding these properties is crucial. They not only help in verifying hyperbolic identities but also provide insights into the behavior of these mathematical constructs.