Problem 47 - Calculus Early Transcendentals: Pearson New International Edition

Question

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

Step-by-Step Solution

Answer

The identity \( \cosh(x+y) = \cosh x \cosh y + \sinh x \sinh y \) is verified to be true.

Step 1: Understand Hyperbolic Functions

Hyperbolic cosine and sine are defined as \( \cosh x = \frac{e^x + e^{-x}}{2} \) and \( \sinh x = \frac{e^x - e^{-x}}{2} \). They are similar to trigonometric functions but relate to the exponential function.

Step 2: Expand the Left Side

Use the definition of hyperbolic cosine for \( \cosh(x+y) \). Plugging into the identity, we have:\[ \cosh(x+y) = \frac{e^{x+y} + e^{-(x+y)}}{2} \].

Step 3: Expand the Right Side

Expand each term in \( \cosh x \cosh y + \sinh x \sinh y \):- \( \cosh x \) is \( \frac{e^x + e^{-x}}{2} \)- \( \cosh y \) is \( \frac{e^y + e^{-y}}{2} \)- \( \sinh x \) is \( \frac{e^x - e^{-x}}{2} \)- \( \sinh y \) is \( \frac{e^y - e^{-y}}{2} \).

Step 4: Multiply and Simplify Right Side Terms

First calculate \( \cosh x \cosh y \):\[ \frac{(e^x + e^{-x})(e^y + e^{-y})}{4} = \frac{e^{x+y} + e^{x-y} + e^{-x+y} + e^{-x-y}}{4} \].Next, calculate \( \sinh x \sinh y \):\[ \frac{(e^x - e^{-x})(e^y - e^{-y})}{4} = \frac{e^{x+y} - e^{x-y} - e^{-x+y} + e^{-x-y}}{4} \].

Step 5: Combine and Simplify Right Side

Add the results from Step 4:\[\frac{1}{4} \left( e^{x+y} + e^{x-y} + e^{-x+y} + e^{-x-y} \right) + \frac{1}{4} \left( e^{x+y} - e^{x-y} - e^{-x+y} + e^{-x-y} \right)\]Combine like terms:\[ \frac{e^{x+y} + e^{-x-y}}{2} \].

Step 6: Verify Identity

Both sides of the original equation simplify to \( \frac{e^{x+y} + e^{-(x+y)}}{2} \), confirming the identity is indeed true:\[ \cosh(x+y) = \cosh x \cosh y + \sinh x \sinh y \].

Key Concepts

Hyperbolic Cosine

The hyperbolic cosine function, denoted as \( \cosh x \), is one of the key hyperbolic functions.It is defined through the exponential function as \( \cosh x = \frac{e^x + e^{-x}}{2} \).
This definition looks similar to the formula for the average value, but it involves an important aspect: the exponential function's growth.The hyperbolic cosine is akin to the ordinary cosine in trigonometry, serving a similar role but focused on hyperbolic angles.
Its values are always positive for all real numbers \( x \), which distinguishes it from the trigonometric cosine.Interestingly, when graphed, \( \cosh x \) produces a smooth curve called a catenary.

• Provides the shape of hanging cables (like electricity lines).
• Forms the curves seen in the design of arches.

In the identity \( \cosh(x+y) = \cosh x \cosh y + \sinh x \sinh y \), \( \cosh x \) and \( \cosh y \) are expanded as products using their respective definitions.This demonstrates how hyperbolic cosine operates additively, a key characteristic in hyperbolic trigonometry.

Hyperbolic Sine

The hyperbolic sine function, represented as \( \sinh x \), is another essential hyperbolic function closely related to \( \cosh x \).
Defined by \( \sinh x = \frac{e^x - e^{-x}}{2} \), it captures the difference between the exponential function and its reciprocal.Similar to the usual sine function in trigonometry, the hyperbolic sine deals with hyperbolic angles.
However, it extends beyond the typical sine since it can produce values greater than 1 directly due to its definition using exponentials.Some attributes of the hyperbolic sine include:

• Odd function, meaning \( \sinh(-x) = -\sinh(x) \).
• Zero value at \( x = 0 \), making it intersect the origin.

The identity \( \cosh(x+y) = \cosh x \cosh y + \sinh x \sinh y \) shows how \( \sinh x \) and \( \sinh y \) contribute additively to the overall expression.By multiplying them, we see how the hyperbolic sine contributes to the identity's whole structure, representing a delicate balance between increase and decrease of exponential terms.

Exponential Function

The exponential function, expressed as \( e^x \), is the backbone of both hyperbolic sine and cosine functions.
It's a mathematical function that quickly grows, making it integral in many fields like calculus, complex analysis, and differential equations.In the realms of hyperbolic functions, its role is foundational.For instance, both \( \cosh x \) and \( \sinh x \) are built from the exponential function \( e^x \) and its reciprocal \( e^{-x} \).Here are key aspects of the exponential function:

• It simplifies many problems in physics, particularly those involving growth and decay.
• Forms the basis of the natural logarithm \( \ln(x) \).
• Plays a pivotal role in creating continuous compounds and population models.

In verifying the identity \( \cosh(x+y) = \cosh x \cosh y + \sinh x \sinh y \), the exponential function connects all parts of both sides.This linkage shows how natural exponents can combine to maintain equalities like hyperbolic identities, reflecting the exponential function's unifying nature in mathematics.