Problem 65 - Calculus Single Variable

Question

Verify the identity \(\cosh (2 x)=2 \cosh ^{2}(x)-1\).

Step-by-Step Solution

Answer

The identity \( \cosh(2x) = 2\cosh^2(x) - 1 \) is verified.

Step 1: Recall the definition of hyperbolic cosine

The hyperbolic cosine function is defined as \( \cosh(x) = \frac{e^x + e^{-x}}{2} \). This definition will be crucial in proving the identity.

Step 2: Express \( \cosh(2x) \) using the definition

Using the definition of the hyperbolic cosine, rewrite \( \cosh(2x) \) as \( \cosh(2x) = \frac{e^{2x} + e^{-2x}}{2} \).

Step 3: Expand \( \cosh^2(x) \)

Recall \( \cosh(x) = \frac{e^x + e^{-x}}{2} \) and square it to get \( \cosh^2(x) = \left( \frac{e^x + e^{-x}}{2} \right)^2 = \frac{e^{2x} + 2 + e^{-2x}}{4} \).

Step 4: Simplify the expression \(2 \cosh^2(x) - 1\)

Calculate \( 2 \cosh^2(x) - 1 \) as follows: multiply \( \frac{e^{2x} + 2 + e^{-2x}}{4} \) by 2 to get \( \frac{e^{2x} + 2 + e^{-2x}}{2} \), then subtract 1 to obtain \( \frac{e^{2x} + 2 + e^{-2x} - 2}{2} = \frac{e^{2x} + e^{-2x}}{2} \).

Step 5: Verify the identity

We found that both \( \cosh(2x) \) and \( 2\cosh^2(x) - 1 \) simplify to \( \frac{e^{2x} + e^{-2x}}{2} \). Therefore, the identity \( \cosh(2x) = 2\cosh^2(x) - 1 \) is verified.

Key Concepts

Hyperbolic Cosine

The hyperbolic cosine, denoted as \( \cosh(x) \), is a function that appears in various mathematical applications, particularly in engineering and physics. It's closely related to the exponential function. The hyperbolic cosine is defined as:

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

In this expression, \( e \) represents the base of natural logarithms, approximately equal to 2.71828. Rather than oscillating like the ordinary cosine function, \( \cosh(x) \) forms a smooth, continuous curve that grows exponentially in the positive and negative directions. It always returns positive values, as it's the average of two exponential terms. This property makes it useful for describing phenomena such as hanging cables or the shape of a suspension bridge, commonly referred to as a catenary curve. For any real number \( x \), the hyperbolic cosine provides a way to model real-world scenarios where balanced forces create natural curves.

Mathematical Identity Verification

Mathematical identities are expressions that are always true for all values of variables involved. They play a vital role in simplifying complex problems. Verifying a mathematical identity involves showing that two different-looking expressions are equivalent. A common approach is to use known definitions, properties, or previously verified identities.In this exercise, we verify the identity \( \cosh(2x) = 2\cosh^2(x) - 1 \). This identity can be proven by substituting the expression for \( \cosh(x) \), and simplifying both sides to see if they match:

• Rewrite \( \cosh(2x) \) using its definition: \( \cosh(2x) = \frac{e^{2x} + e^{-2x}}{2} \).
• Calculate \( \cosh^2(x) \) and manipulate it to match the expression of \( \cosh(2x) \).

Through manipulation and simplification, one confirms that both sides equal the same expression, verifying the identity. This process is crucial in mathematical proofs and problem-solving, helping simplify and solve equations.

Exponential Functions

Exponential functions are among the most important mathematical functions, characterized by a constant base raised to a variable exponent. The usual base for these functions is \( e \), known as Euler's number:

• \( f(x) = e^x \)

Exponential functions are essential because they model growth and decay processes, like populations, radioactive decay, and interest calculations. They exhibit unique properties such as:

• Rapid increase or decrease with positive or negative exponents.
• Never equaling zero, remaining always positive.
• The derivative of \( e^x \) is itself, \( \frac{d}{dx}(e^x) = e^x \), which simplifies analysis.

Hyperbolic functions, like \( \cosh(x) \), are defined using exponential functions. These definitions allow translating problems involving hyperbolic functions into those involving exponential expressions. Understanding exponential behavior is fundamental, assisting in comprehending how functions like \( \cosh(x) \) and related identities work.