Problem 11 - Single Variable Calculus

Question

Prove the identity. \(\cosh ^{2} x=\frac{1+\cosh 2 x}{2}\)

Step-by-Step Solution

Answer

Using the definition of hyperbolic cosine and sine functions (\(\cosh x = \frac{e^x + e^{-x}}{2}\) and \(\sinh x = \frac{e^x - e^{-x}}{2}\)), we compute \(\cosh^2 x = \frac{e^{2x} + 2 + e^{-2x}}{4}\) and \(\sinh^2 x = \frac{e^{2x} - 2 + e^{-2x}}{4}\). Then, using the double-angle formula for hyperbolic cosine (\(\cosh (2x) = \cosh^2 x - \sinh^2 x\)), we find \(\cosh (2x) = 1\). Plugging this expression into the identity we want to prove, we see that \(\frac{1+\cosh(2x)}{2} = \frac{1 + 1}{2} = \frac{2}{2} = 1\), which confirms that \(\cosh^2 x = \frac{1+\cosh(2x)}{2}\).

Step 1: Recall the definition of the hyperbolic cosine function

The hyperbolic cosine function is defined as \(\cosh x = \frac{e^x + e^{-x}}{2}\). We will use this definition to find \(\cosh^2 x\).

Step 2: Compute \(\cosh^2 x\)

Using the definition of the hyperbolic cosine function, we can find \(\cosh^2 x\) as follows: \(\cosh^2 x = \left(\frac{e^x + e^{-x}}{2}\right)^2 = \frac{e^{2x} + 2 + e^{-2x}}{4}\)

Step 3: Recall the double-angle formula for hyperbolic cosine

The double-angle formula for the hyperbolic cosine function is given by \(\cosh (2x) = \cosh^2 x - \sinh^2 x\). To make use of this formula, we need to find an expression for \(\sinh^2 x\).

Step 4: Recall the definition of the hyperbolic sine function and find an expression for \(\sinh^2 x\)

The hyperbolic sine function is defined as \(\sinh x = \frac{e^x - e^{-x}}{2}\). Using this definition, we can find \(\sinh^2 x\) as follows: \(\sinh^2 x = \left(\frac{e^x - e^{-x}}{2}\right)^2 = \frac{e^{2x} - 2 + e^{-2x}}{4}\)

Step 5: Use the double-angle formula to find an expression for \(\cosh (2x)\)

Now we can use the double-angle formula to find an expression for \(\cosh (2x)\): \(\cosh (2x) = \cosh^2 x - \sinh^2 x = \left(\frac{e^{2x} + 2 + e^{-2x}}{4}\right) - \left(\frac{e^{2x} - 2 + e^{-2x}}{4}\right)\) This simplifies to: \(\cosh (2x) = \frac{4}{4} = 1\)

Step 6: Plug the expression for \(\cosh (2x)\) into the identity we want to prove

Finally, let's plug the expression we found for \(\cosh (2x)\) into the identity we want to prove: \(\frac{1+\cosh(2x)}{2} = \frac{1 + 1}{2} = \frac{2}{2} = 1\) Since we've found that \(\cosh^2 x = 1\) and \(\frac{1+\cosh(2x)}{2} = 1\), the identity \(\cosh^2 x = \frac{1+\cosh(2x)}{2}\) has been proven.

Key Concepts

Hyperbolic Cosine

The hyperbolic cosine, denoted as \(\cosh x\), is a fundamental function in hyperbolic trigonometry. Similar to its circular counterpart, the cosine function, it is related to the exponential function. The hyperbolic cosine is defined by the formula:

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

This formula captures the essence of the function by taking the average of exponential growth and decay. Understanding \(\cosh x\) is crucial when dealing with hyperbolic identities or transformations. It describes how a value changes across hyperbolic planes. Using this definition, you can calculate squares of hyperbolic cosines to solve exercises involving squared identities.

Hyperbolic Sine

Hyperbolic sine, represented as \(\sinh x\), forms the complement to hyperbolic cosine in hyperbolic trigonometry. The relationship between \(\cosh x\) and \(\sinh x\) helps in solving various identities. The definition is given by:

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

This expression highlights the difference between exponential expansion and contraction, a key characteristic of hyperbolic functions. To compute \(\sinh^2 x\), simply square the expression, leading to:

• \(\sinh^2 x = \left(\frac{e^x - e^{-x}}{2}\right)^2 = \frac{e^{2x} - 2 + e^{-2x}}{4}\)

Familiarity with \(\sinh x\) and its square is necessary to apply and prove identities in calculus and complex analysis contexts.

Double-angle Formulas

Double-angle formulas are vital tools in mathematical problem-solving, especially when working with trigonometric and hyperbolic identities. For hyperbolic functions, one such formula involves \(\cosh (2x)\):

• \(\cosh (2x) = \cosh^2 x - \sinh^2 x\)

This formula connects the values of \(\cosh\) and \(\sinh\) in such a way that it simplifies calculations involving double angles. For our exercise, this step is crucial to derive expressions that simplify the identity given. Understanding this formula allows one to transition smoothly from single to double angles when proving or manipulating hyperbolic equations.

Trigonometric Identities

Trigonometric identities serve as the backbone for numerous mathematical proofs and simplifications. While often associated with circular functions, hyperbolic trigonometric identities share a similar significance. They allow us to express complex relationships between functions like \(\cosh x\) and \(\sinh x\) in simpler forms. When faced with an identity such as \(\cosh^2 x = \frac{1+\cosh 2x}{2}\), utilizing these identities helps verify and prove the equation effectively:

• Begin by expressing \(\cosh^2 x\) and \(\sinh^2 x\) using their definitions.
• Apply the double-angle formulas to transform expressions into recognisable identities.
• Simplify and check equivalence on both sides of the identity.

Mastery of these identities enables you to navigate complex problems with greater ease and derive results that might seem initially inaccessible.