Problem 18 - Calculus Early Transcendentals

Question

Use the given identity to verify the related identity. Use the identity \(\cosh (x+y)=\cosh x \cosh y+\sinh x \sinh y\) to verify the identity \(\cosh 2 x=\cosh ^{2} x+\sinh ^{2} x.\)

Step-by-Step Solution

Answer

Question: Verify the identity \(\cosh 2x = \cosh^2 x + \sinh^2 x\). Answer: Using the identity \(\cosh(x+y) = \cosh x\cosh y + \sinh x\sinh y\), we substituted \(x\) with \(2x\), and found that \(\cosh(2x) = \cosh^2 x + \sinh^2 x\), which verifies the given identity.

Step 1: Substitute \(x\) with \(2x\)

First, we need to substitute \(x\) with \(2x\) in the given identity to get the result we want. We have, \(\cosh(x+y) = \cosh x\cosh y + \sinh x\sinh y\) Now, let us substitute \(y = x\): \(\cosh(2x) = \cosh x\cosh x + \sinh x\sinh x\)

Step 2: Simplify the equation

Now, we have: \(\cosh(2x) = \cosh^2 x + \sinh^2 x\) This means that we have verified the identity \(\cosh 2x = \cosh^2 x + \sinh^2 x\).

Key Concepts

Understanding the Cosh Function

The hyperbolic cosine function, often written as \( \cosh(x) \), is a fundamental component in hyperbolic trigonometry. It is analogous to the cosine function in circular trigonometry, but adapted for the hyperbola.
The definition of the \( \cosh \) function is quite analytical. It can be expressed using the exponential function: \[ \cosh(x) = \frac{e^x + e^{-x}}{2} \] This definition highlights its key property: \( \cosh(x) \) always produces a value greater than or equal to 1 for all real numbers \( x \).
Here are a few crucial facts about \( \cosh \):

• \( \cosh(0) = 1 \)
• It is even, meaning \( \cosh(-x) = \cosh(x) \)
• Increases exponentially as \( x \) moves away from zero in both directions

Moreover, the \( \cosh \) function is often applied in physical contexts, such as calculating the shape of a hanging cable or chain, known as the catenary. Understanding \( \cosh \) is essential for anyone diving into hyperbolic identities.

Exploring the Sinh Function

The hyperbolic sine function, denoted \( \sinh(x) \), is the counterpart to the hyperbolic cosine function. While \( \cosh(x) \) relates to the average of exponentials, \( \sinh(x) \) can be expressed as the difference between two exponentials: \[ \sinh(x) = \frac{e^x - e^{-x}}{2} \] This definition demonstrates that the hyperbolic sine generates a wide range of real numbers, including both positive and negative values.
Some important properties of \( \sinh \) include:

• \( \sinh(0) = 0 \)
• It is an odd function, meaning \( \sinh(-x) = -\sinh(x) \)
• Shows exponential growth in magnitude as \( x \) moves away from zero

Unlike trigonometric sine, which cycles between -1 and 1, \( \sinh \) exhibits unbounded behavior.
Understanding \( \sinh \) helps with analyzing hyperbolic identities, as it often pairs with \( \cosh \) to form important identities used in calculus and engineering applications.

Identity Verification

Identity verification in mathematics involves proving that two expressions are equivalent, using established identities or transformation techniques. This process ensures the reliability and consistency of the mathematical statements we use.
In the context of hyperbolic functions, verifying identities like \[ \cosh(2x) = \cosh^2(x) + \sinh^2(x) \] involves substituting values and simplifying expressions to check their equivalence using known identities.
The identity provided here utilizes the formula: \[ \cosh(x+y) = \cosh x \cosh y + \sinh x \sinh y \] In this scenario, substituting \( y = x \) immediately confirms the target identity. Practicing identity verification strengthens understanding by reinforcing connections between different mathematical concepts.
Through this approach, students can develop a deeper insight into why these identities work and how they interconnect with larger mathematical frameworks. It's more than just proving equations—it's about understanding the inherent properties of the functions involved.