Problem 64 - Calculus Single Variable

Question

Verify the identity \(\sinh (x+y)=\sinh (x) \cosh (y)+\) \(\cosh (x) \sinh (y)\).

Step-by-Step Solution

Answer

The identity is verified; both sides simplify to \( \frac{e^{x+y} - e^{-(x+y)}}{2} \).

Step 1: Recall Definitions

Recall the definitions of hyperbolic sine and cosine functions. They are given by: \( \sinh(x) = \frac{e^x - e^{-x}}{2} \) and \( \cosh(x) = \frac{e^x + e^{-x}}{2} \).

Step 2: Write Left Side

Substitute the hyperbolic definitions into the left side of the identity: \( \sinh(x+y) = \frac{e^{x+y} - e^{-(x+y)}}{2} \).

Step 3: Express Exponentials

Break the compound exponentials \( e^{x+y} \) and \( e^{-(x+y)} \) into products: \( e^{x+y} = e^x e^y \) and \( e^{-(x+y)} = e^{-x} e^{-y} \).

Step 4: Substitute Into Identity

Replace these expressions back into the formula for \( \sinh(x+y) \): \( \sinh(x+y) = \frac{e^x e^y - e^{-x} e^{-y}}{2} \).

Step 5: Write Right Side

Substitute the definitions for \( \sinh(x) \), \( \cosh(y) \), \( \cosh(x) \), and \( \sinh(y) \): \( \sinh(x) \cosh(y) = \frac{(e^x - e^{-x})(e^y + e^{-y})}{4} \), \( \cosh(x) \sinh(y) = \frac{(e^x + e^{-x})(e^y - e^{-y})}{4} \).

Step 6: Expand Products

Expand the products from Step 5: \( \sinh(x) \cosh(y) = \frac{e^{x+y} + e^{x-y} - e^{-x+y} - e^{-(x+y)}}{4} \) and \( \cosh(x) \sinh(y) = \frac{e^{x+y} - e^{x-y} + e^{-x+y} - e^{-(x+y)}}{4} \).

Step 7: Combine Right Side Terms

Add the two expanded products from Step 6: \( \sinh(x) \cosh(y) + \cosh(x) \sinh(y) = \frac{2e^{x+y} - 2e^{-(x+y)}}{4} = \frac{e^{x+y} - e^{-(x+y)}}{2} \).

Step 8: Verify Equality

Both the left side and right side simplify to \( \frac{e^{x+y} - e^{-(x+y)}}{2} \), thus verifying that the identity \( \sinh(x+y) = \sinh(x) \cosh(y) + \cosh(x) \sinh(y) \) is true.

Key Concepts

Hyperbolic Sine

The hyperbolic sine function, often denoted as \( \sinh(x) \), is one of the fundamental hyperbolic functions used in mathematics, particularly in areas involving hyperbolic geometry and complex analysis. It is defined using the exponential function:

• The formula for hyperbolic sine is \( \sinh(x) = \frac{e^x - e^{-x}}{2} \).
• The graph of \( \sinh(x) \) resembles a regular sine wave, but with crucial differences in its growth behavior at positive and negative extremes.

These functions have a unique property where they are related to exponential growth and decay. They play a crucial role in defining hyperbolas and have applications in calculus and differential equations. Understanding \( \sinh(x) \) involves recognizing its behavior in comparison to the regular sine function and noting how it spans the y-axis with a noticeable zero crossing point at the origin. The symmetry and analyticity of \( \sinh(x) \) make it a powerful tool for solving problems in hyperbolic trigonometry.

Hyperbolic Cosine

The hyperbolic cosine function, represented by \( \cosh(x) \), complements \( \sinh(x) \) within the family of hyperbolic functions. It is defined similarly, using the exponential function:

• Its formula is \( \cosh(x) = \frac{e^x + e^{-x}}{2} \).
• The curve formed by \( \cosh(x) \) is known for its bell-like shape, unlike the waving structure of its hyperbolic sine counterpart.

One of the fascinating properties of \( \cosh(x) \) is its relationship to even and odd functions. As an even function, \( \cosh(x) \) is symmetric about the y-axis, much like the regular cosine function. This characteristic allows it to retain its values when the input is negated, evident from \( \cosh(-x) = \cosh(x) \). The hyperbolic cosine is particularly important in physics and engineering, where it describes the equation of a hanging cable or chain, known as a catenary. By grasping \( \cosh(x) \), students can better understand the equations governing natural phenomena and the underlying symmetry of exponential functions in hyperbolic contexts.

Verification of Identities

Verifying mathematical identities is an essential skill in algebra and calculus that strengthens comprehension and problem-solving skills. When we verify an identity, we essentially demonstrate that two expressions are equivalent under certain conditions.

• For hyperbolic functions, such as in the problem \( \sinh(x+y) = \sinh(x) \cosh(y) + \cosh(x) \sinh(y) \), verification confirms their reliability and application.
• To verify an identity, one typically transforms one side into the other, showing that both yield the same result.

In the given exercise, this process involves substituting the definitions of \( \sinh \) and \( \cosh \) into each side of the equation and confirming they produce the same simplified form. This procedure often involves breaking down complex exponentials using known identities, expanding products, simplifying fractions, and using symmetry properties of the functions involved. Mastering the verification of identities, like our hyperbolic example, not only confirms mathematical principles but also provides insight into the structural elegance and interconnectedness of mathematical concepts.