Problem 52 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 41-52, verify that the given equations are identities. \(\cosh 2 x=\cosh ^{2} x+\sinh ^{2} x\)

Step-by-Step Solution

Answer

The identity \( \cosh 2x = \cosh^2 x + \sinh^2 x \) is verified using the hyperbolic double angle formula.

Step 1: Recall Hyperbolic Identities

Recall that the hyperbolic cosine and hyperbolic sine identities are similar to the Pythagorean identities. For hyperbolic functions, we have \( \cosh^2 x - \sinh^2 x = 1 \). This will help us verify the given identity.

Step 2: Use Hyperbolic Double Angle Formula

Recall the hyperbolic double angle formula for cosine, which states \( \cosh(2x) = \cosh^2 x + \sinh^2 x \). This is consistent with the identity to be verified.

Step 3: Simplify and Verify the Identity

Recognize that \( \cosh(2x) \) is defined as \( \cosh^2 x + \sinh^2 x \) due to the hyperbolic double angle identity. This confirms that the original given equation is indeed an identity, as both sides are equal.

Key Concepts

Hyperbolic Identities

Hyperbolic identities work much like the familiar trigonometric identities, such as the Pythagorean identity. The primary hyperbolic identities are founded on hyperbolic functions such as hyperbolic sine (\(\sinh\)) and hyperbolic cosine (\(\cosh\)). For these functions, a key identity is

• \(\cosh^2 x - \sinh^2 x = 1\)

This identity resembles the trigonometric identity \(\cos^2 x + \sin^2 x = 1\) but note the negative sign between \(\cosh^2 x\) and \(\sinh^2 x\). This equation arises from the definition of hyperbolic functions, where \(\sinh x = \frac{e^x - e^{-x}}{2}\) and \(\cosh x = \frac{e^x + e^{-x}}{2}\). Both functions lend themselves to various simplifications in calculations involving hyperbolic expressions.Hyperbolic identities simplify the analysis of hyperbolic functions, just as trigonometric identities do for trigonometric functions. Mastering these will be crucial for solving calculus problems or verifying hyperbolic identities swiftly.

Double Angle Formulas

Double angle formulas are an essential part of manipulating both trigonometric and hyperbolic functions in calculus. For hyperbolic functions, the double angle identity helps relate the function at \(2x\) to the function at \(x\).The double angle formulas for hyperbolic functions include:

• \(\cosh(2x) = \cosh^2 x + \sinh^2 x \)
• \(\sinh(2x) = 2\sinh x \cosh x \)

The formula \(\cosh(2x) = \cosh^2 x + \sinh^2 x \) allows for the simplification and verification of hyperbolic identities, especially when comparing the structure of expressions involving hyperbolic functions. These formulas enable swift manipulation and transformation of expressions in calculus problems.In practice, these formulas allow the reduction of complex hyperbolic expressions, making it easier to compute integrals, derivatives, or when solving equations. Fully understanding these formulas enables seamless and efficient problem-solving.

Calculus Problems

Calculus problems involving hyperbolic functions often require the use of their identities and specific formulas. When working with problems, you may need to perform tasks such as differentiation or integration involving \(\sinh x\) and \(\cosh x\).A typical problem-solving approach includes:

• Recognizing or rewriting expressions using hyperbolic identities
• Applying double angle or other specific formulas
• Simplifying expressions to find derivatives or integrals

For example, recognizing \(\cosh 2x = \cosh^2 x + \sinh^2 x\) allows a seamless transition into solving more complex calculus expressions.Consider the derivative of \(\cosh x\), which is \(\sinh x\), and the derivative of \(\sinh x\), which is \(\cosh x\). A unique feature is that both derivatives are closely linked to each other and can also help when dealing with integration problems.Mastery of these basic calculus techniques involving hyperbolic functions is key to solving not just textbook problems but also real-world applications they model.