Question

Verify that the given equations are identities. \(e^{x}=\cosh x+\sinh x\)

Short answer

The equation is an identity: \( e^x = \cosh x + \sinh x \).

Step-by-step solution

Recall the Definitions of Hyperbolic Functions

The hyperbolic cosine function is defined as: \[ \cosh x = \frac{e^x + e^{-x}}{2} \] and the hyperbolic sine function is defined as: \[ \sinh x = \frac{e^x - e^{-x}}{2} \] These definitions will be used in the verification of the identity.

Substitute into the Right Side of the Identity

Substitute the definitions of \( \cosh x \) and \( \sinh x \) into the right side of the identity \( \cosh x + \sinh x \): \[ \cosh x + \sinh x = \left( \frac{e^x + e^{-x}}{2} \right) + \left( \frac{e^x - e^{-x}}{2} \right) \]

Simplify the Expression

Combine the fractions by adding the numerators: \[ \frac{e^x + e^{-x} + e^x - e^{-x}}{2} = \frac{2e^x}{2} \] Simplify the fraction: \[ e^x \]

Compare Both Sides of the Equation

Both sides of the original equation are \( e^x \). Therefore, the left side \( e^x \) is equal to the simplified right side \( e^x \). This verifies that the equation \( e^x = \cosh x + \sinh x \) is an identity.

Key concepts

Mathematical Identities

Mathematical identities are equations that hold true for all values of the variables involved. In simpler terms, they represent universal truths within mathematics, where each side of the equation is equivalent no matter what numbers are plugged in. Understanding identities is fundamental as it allows for simplifying complex equations and verifying relationships among functions. For hyperbolic functions, identities often involve recognizing patterns that always remain consistent.

• For instance, in the equation we're examining \(e^x = \cosh x + \sinh x\), both sides are identical for any real number \(x\).


• These identities help build more complex mathematical concepts. They serve as foundational tools in calculus and differential equations.

Identifying and using these identities effectively can simplify problem solving and provide deeper insights into the relationships between various mathematical functions.

Exponential Functions

Exponential functions are a vital part of mathematics, commonly written in the form \(e^x\), where \(e\) is the base of the natural logarithm, approximately equal to 2.71828. A noteworthy feature of exponential functions is their growth rate; they increase rapidly and are continuous and differentiable.

• Exponential functions occur naturally in many areas of science, such as in describing populations, in radioactive decay, and even in finance.


• They possess unique properties that make them useful in defining hyperbolic functions like \(\cosh x\) and \(\sinh x\).


• Recognizing how exponential functions intertwine with hyperbolic functions aids in simplifying and solving complex equations.

Understanding the behaviors and properties of exponential functions assists in decoding and verifying mathematical identities involving hyperbolic functions.

Equation Simplification

Simplifying an equation often involves reducing its complexity while preserving equality. In our verification exercise, we simplified the expression \(\cosh x + \sinh x\) to show its equivalence to \(e^x\). This is achieved by recognizing and applying the properties of hyperbolic and exponential functions.

• Substitution of the hyperbolic function definitions into the equation transforms a complex expression into a simpler form.


• Combining terms and eliminating contradictions allows for clarity, helping us to unveil relationships within the equation.


• Simplification also involves algebraic manipulations such as combining like terms, handling fractions, and canceling terms.

By mastering simplification techniques, mathematical identities and relationships are more easily proven, and solutions are more readily obtained.

Verification of Identities

Verification of identities involves showing that two expressions are indeed equivalent by logical steps and mathematical principles. In this exercise, we validated the identity \(e^x=\cosh x+\sinh x\). This process starts with understanding both sides of the equation.

• We used known identities, like the definitions of \(\cosh\) and \(\sinh\), and substituted them into one side of the equation.


• Upon simplifying, we showed that both sides of the equation resolve to the same expression.


• Verification often involves checking for consistency across different values or types of inputs.

This process provides confidence in mathematical principles, essential for ensuring that theories are well-founded and applicable.