Problem 13 - Calculus Early Transcendentals

Question

Verify each identity using the definitions of the hyperbolic functions. \(\cosh 2 x=\cosh ^{2} x+\sinh ^{2} x\) (Hint: Begin with the right side of the equation.)

Step-by-Step Solution

Answer

Question: Verify the identity \(\cosh 2 x=\cosh ^{2} x+\sinh ^{2} x\) using the definitions of the hyperbolic functions. Answer: Starting with the right side of the equation, we substitute the definitions of the hyperbolic functions and simplify the terms. We find that the right side equals \(\frac{e^{2x} + e^{-2x}}{2}\), which is the definition of \(\cosh 2x\). Thus, we have verified the identity \(\cosh 2 x=\cosh ^{2} x+\sinh ^{2} x\).

Step 1: Identify the definitions of the hyperbolic functions

The hyperbolic cosine and hyperbolic sine are defined as: \(\cosh x = \frac{e^x + e^{-x}}{2}\) and \(\sinh x = \frac{e^x - e^{-x}}{2}.\)

Step 2: Substitute the definitions into the right side of the equation

We will now substitute the definitions of \(\cosh x\) and \(\sinh x\) into the right side of the equation: \((\cosh ^{2} x+\sinh ^{2} x) = \left(\frac{e^x + e^{-x}}{2}\right)^2 + \left(\frac{e^x - e^{-x}}{2}\right)^2.\)

Step 3: Simplify the terms

Let's simplify the squared terms in the equation: \(\left(\frac{e^x + e^{-x}}{2}\right)^2 + \left(\frac{e^x - e^{-x}}{2}\right)^2 = \frac{(e^x + e^{-x})^2}{4} + \frac{(e^x - e^{-x})^2}{4}.\)

Step 4: Combine the fractions

Now, let's combine the fractions under a common denominator: \(\frac{(e^x + e^{-x})^2}{4} + \frac{(e^x - e^{-x})^2}{4} = \frac{(e^x + e^{-x})^2 + (e^x - e^{-x})^2}{4}.\)

Step 5: Expand the numerator

Next, we will expand the squared terms in the numerator: \(\frac{(e^{2x} + 2e^x e^{-x} + e^{-2x}) + (e^{2x} - 2e^x e^{-x} + e^{-2x})}{4}.\)

Step 6: Simplify the numerator

We can now simplify the numerator by combining like terms: \(\frac{2e^{2x} + 2e^{-2x}}{4}.\)

Step 7: Simplify further

We can simplify the expression further by dividing each term by 2: \(\frac{e^{2x} + e^{-2x}}{2}.\)

Step 8: Recognize the hyperbolic cosine

Finally, we notice that the simplified expression is the definition of \(\cosh 2x\): \(\cosh 2x = \frac{e^{2x} + e^{-2x}}{2}.\) Now that we have derived the left side of the equation, we have verified the identity: \(\cosh 2 x=\cosh ^{2} x+\sinh ^{2} x\).

Key Concepts

Hyperbolic Functions

Hyperbolic functions are analogs of the ordinary trigonometric, or circular, functions but they are designed to describe hyperbolas instead of circles. They are especially useful in various areas of mathematics and engineering, particularly in complex analysis and hyperbolic geometry.
To make sense of hyperbolic functions, consider:

• The hyperbolic sine, denoted as \( \sinh x \), is related to the exponential function and can be visualized as half the difference between two exponential functions.
• The hyperbolic cosine, \( \cosh x \), similarly corresponds to an expression involving two exponential functions, but in this case, it is the half sum.

Hyperbolic functions also find their similarities in calculus, where they exhibit properties akin to trigonometric functions. However, unlike trigonometric identities which revolve around angles in a circle, hyperbolic identities hinge on properties of hyperbolas. That is how these unique functions contribute to simplifying and solving complex mathematical equations.

Cosh and Sinh Definitions

Understanding the definitions of \( \cosh x \) and \( \sinh x \) is crucial for working with hyperbolic identities. These definitions are straightforward:

• \( \cosh x = \frac{e^x + e^{-x}}{2} \)
• \( \sinh x = \frac{e^x - e^{-x}}{2} \)

Why base these functions on exponential expressions? Well, exponentials are fundamental to mathematics due to their simple derivative properties. Therefore, hyperbolic functions also enjoy similar properties, making calculus operations on them quite manageable.
In practical computation, knowing these definitions helps us substitute these expressions accurately into various identities or functions. For instance, verifying the identity \( \cosh 2x = \cosh^2 x + \sinh^2 x \) depends on correctly applying these initial definitions and manipulating the expressions they provide.

Identity Verification

Identity verification in mathematics involves confirming that two expressions, initially appearing different, are indeed equivalent by transformation using baselines like definitions and properties.
In the case of hyperbolic functions, verifying identities often involves a step-by-step substitution and simplification process. Take the verification of \( \cosh 2x = \cosh^2 x + \sinh^2 x \) as an example:

• Start with the right-hand side, substituting the definitions of \( \cosh x \) and \( \sinh x \).
• Simplify the resulting algebraic expression using algebraic identities like expanding square terms and combining like terms correctly.
• Finally, observe that the simplified outcome matches the left-hand function form, confirming the identity.

This verification doesn't just confirm equality but also deepens understanding of the underlying structures. Successfully completing these steps is essential for mastery over principles bringing cohesion and coherence to different areas in mathematics.