Question

Prove \(\sinh (x+y)=\sinh (x) \cosh (y)+\cosh (x) \sinh (y)\) by changing the expression to exponentials.

Short answer

Equation is proven using exponential definitions and simplification.

Step-by-step solution

Recall Definitions of Hyperbolic Functions

The function \( \sinh(x) \) is defined as \( \sinh(x) = \frac{e^x - e^{-x}}{2} \), and the function \( \cosh(x) \) is defined as \( \cosh(x) = \frac{e^x + e^{-x}}{2} \).

Write \( \sinh(x+y) \) Using Exponentials

Using the definition of hyperbolic sine, write:\[ \sinh(x+y) = \frac{e^{x+y} - e^{-(x+y)}}{2} \]

Expand the Exponential Terms

Expand both \( e^{x+y} \) and \( e^{-(x+y)} \): - \( e^{x+y} = e^x e^y \) - \( e^{-(x+y)} = e^{-x} e^{-y} \)So, the expression becomes:\[ \sinh(x+y) = \frac{e^x e^y - e^{-x} e^{-y}}{2} \]

Apply Definitions to the Right-Hand Side

Substitute the exponential terms into the right-hand side of the identity to be proven:\[ \sinh(x) \cosh(y) + \cosh(x) \sinh(y) = \left(\frac{e^x - e^{-x}}{2}\right)\left(\frac{e^y + e^{-y}}{2}\right) + \left(\frac{e^x + e^{-x}}{2}\right)\left(\frac{e^y - e^{-y}}{2}\right) \]

Simplify the Right-Hand Side

Multiply the terms:For the first product:\[ \frac{(e^x - e^{-x})(e^y + e^{-y})}{4} = \frac{e^{x+y} + e^{y-x} - e^{-x+y} - e^{-(x+y)}}{4} \]For the second product:\[ \frac{(e^x + e^{-x})(e^y - e^{-y})}{4} = \frac{e^{x+y} - e^{y-x} + e^{-x+y} - e^{-(x+y)}}{4} \]

Combine and Simplify Equations

Add the results from Step 5:Combine like terms:\[ \frac{e^{x+y} - e^{-(x+y)}}{2} \]This is equal to \( \sinh(x+y) \) as found in Step 2.

Key concepts

Hyperbolic Functions

Hyperbolic functions are analogues of trigonometric functions but for a hyperbola rather than a circle. Just as trigonometric functions like sine and cosine are based on the relationships of angles within a unit circle, hyperbolic sine (\( \sinh \)) and hyperbolic cosine (\( \cosh \)) relate to a unit hyperbola.
These functions are essential in various mathematical applications, including calculus and differential equations. They can model real-world phenomena involving exponential growth and decay.

• Hyperbolic Sine: The function \( \sinh(x) \) is defined as \( \sinh(x) = \frac{e^x - e^{-x}}{2} \). It represents how much a point on a hyperbola is off the horizontal axis.


• Hyperbolic Cosine: The function \( \cosh(x) \) is defined as \( \cosh(x) = \frac{e^x + e^{-x}}{2} \). It reflects how far the point is from the vertical axis.

These definitions make hyperbolic functions intimately connected to exponential functions, which are crucial for proving identities and solving equations.

Exponential Functions

Exponential functions form the backbone of hyperbolic functions and are given by equations involving powers of the constant \( e \), approximately 2.718. They are characterized by their rapid growth or decay, making them vital in fields ranging from compound interest calculations to population dynamics.

• Definition: A basic exponential function looks like \( e^x \), where \( e \) is the base of the natural logarithm and \( x \) is the exponent.


• Properties: Exponential functions have unique properties, such as \( e^{x+y} = e^x \cdot e^y \) and \( e^{-x} = \frac{1}{e^x} \). These properties are leveraged to expand hyperbolic functions in terms of exponentials.

In proving the hyperbolic identity, we express \( \sinh \) and \( \cosh \) as combinations of exponential functions, enabling simplification and proof through algebraic manipulation.

Mathematical Proofs

Mathematical proofs are rigorous arguments used to demonstrate the truth of mathematical statements. They require sound logic and understanding of fundamental concepts.
In the case of hyperbolic identities, proofs often involve substituting definitions and simplifying expressions until both sides of an equation match.

• **Structure of Proofs:** A typical proof consists of introducing definitions, logical deductions, and algebraic simplifications. It may also involve substituting known identities to simplify the problem.


• **Simplification Techniques:** By working with the definitions of \( \sinh(x) \) and \( \cosh(y) \) in terms of exponentials, we simplify the problem from a purely abstract identity to a series of algebraic steps.

This type of proof not only verifies a mathematical statement but also helps in developing a deeper understanding of the relationships between exponential and hyperbolic functions. It nurtures analytical thinking, which is integral for higher-level mathematics.