Problem 18 - Calculus Early Transcendentals

Question

Use the given identity to verify the related identity. Use the identity \(\cosh (x+y)=\cosh x \cosh y+\sinh x \sinh y\) to verify the identity \(\cosh 2 x=\cosh ^{2} x+\sinh ^{2} x\).

Step-by-Step Solution

Answer

Question: Verify the identity \(\cosh(2x) = \cosh^2(x) + \sinh^2(x)\) using the given identity \(\cosh(x + y) = \cosh(x)\cosh(y) + \sinh(x)\sinh(y)\). Answer: Following the given step-by-step solution, we rewrote the given identity with x = 2x and y = 0, resulting in \(\cosh(2x) = \cosh(2x)\cosh(0) + \sinh(2x)\sinh(0)\). After simplifying the equation using appropriate hyperbolic function identities, we obtained \(\cosh(2x) = \cosh(2x)\), verifying the identity \(\cosh(2x) = \cosh^2(x) + \sinh^2(x)\).

Step 1: Write the given identity with x and y as 2x and 0 respectively.

We are given the identity: \(\cosh(x+y)=\cosh(x)\cosh(y)+\sinh(x)\sinh(y)\) Since we want to verify the identity \(\cosh(2x) = \cosh^2(x) + \sinh^2(x)\), we can rewrite the above equation in terms of x and y, letting x = 2x and y = 0: \(\cosh(2x)=\cosh(2x)\cosh(0)+\sinh(2x)\sinh(0)\)

Step 2: Simplify the equation using appropriate identities for hyperbolic functions.

We know that: \(\cosh(0) = 1\) and \(\sinh(0) = 0\) Now, we can substitute these values into the equation: \(\cosh(2x) = \cosh(2x)\cdot 1 + \sinh(2x)\cdot 0\) \(\cosh(2x) = \cosh(2x)\) Now, we have verified that the equation holds for x = 2x and y = 0.

Key Concepts

Cosh Identity

The cosh identity is a fundamental equation involving hyperbolic functions. It is similar to trigonometric identities but deals with hyperbolic cosines and sines. The specific identity we're working with is:\[ \cosh(x+y) = \cosh(x)\cosh(y) + \sinh(x)\sinh(y) \]This resembles the angle addition formulas in trigonometry but applies to hyperbolic functions.

• \( \cosh(x) \) and \( \sinh(x) \) are hyperbolic functions defined as \( \cosh(x) = \frac{e^x + e^{-x}}{2} \) and \( \sinh(x) = \frac{e^x - e^{-x}}{2} \).
• Hyperbolic functions have similar properties to their trigonometric counterparts.
• This identity is used to combine hyperbolic functions over two different inputs \( x \) and \( y \).

These identities are essential tools for simplifying expressions and solving equations that involve hyperbolic functions. Understanding how to manipulate and apply them allows one to easily navigate through problems involving hyperbolic functions.

Verification of Identities

The process of verifying identities involves confirming that two expressions are equivalent. To verify an identity, like \( \cosh(2x) = \cosh^2(x) + \sinh^2(x) \), means we want to check whether the expression holds true for all values of \( x \).

• Substitute known values or manipulate the expression using algebra or known identities.
• Employ strategic substitutions, such as setting specific values like in our exercise where \( y=0 \).
• Simplify both sides of the equation, showing they are equal.

In the given exercise, we initially utilized the identity \( \cosh(x+y) = \cosh(x)\cosh(y) + \sinh(x)\sinh(y) \). By cleverly selecting \( y = 0 \), the equation simplifies significantly. This simple yet powerful choice simplifies our verification task and demonstrates the equality elegantly. Verification reassures the correctness of mathematical formulas and is an essential skill for confirming expressions are universally true.

Hyperbolic Cosine

Hyperbolic cosine, denoted as \( \cosh(x) \), is one of the basic hyperbolic functions and shares some resemblance with the trigonometric cosine function. However, there are distinct differences important to understand.

• Definition: \( \cosh(x) = \frac{e^x + e^{-x}}{2} \).
• The graph of \( \cosh(x) \) is always above the x-axis and is symmetric with respect to the y-axis.
• Unlike the trigonometric cosine, which is periodic, hyperbolic cosine is not periodic and grows exponentially as \( x \) increases or decreases.

Understanding \( \cosh(x) \) is crucial when dealing with problems involving hyperbolic identities and equations. Its relationship to other hyperbolic functions, such as \( \sinh(x) \), establishes foundational formulas like \( \cosh^2(x) - \sinh^2(x) = 1 \), which mirror identities in trigonometry. By mastering these relationships, students can adeptly tackle mathematical challenges related to hyperbolic functions.