Problem 71 - College Algebra

Question

The hyperbolic cosine and hyperbolic sine functions are defined by $$\cosh x=\frac{e^{x}+e^{-x}}{2} \quad \text { and } \quad \sinh x=\frac{e^{x}-e^{-x}}{2}$$ Prove that \((\cosh x)^{2}-(\sinh x)^{2}=1\).

Step-by-Step Solution

Answer

After substituting the definitions of \cosh(x) and \sinh(x), simplifying and checking the value, we find that indeed, \((\cosh x)^{2} - (\sinh x)^{2}\) equals to 1.

Step 1: Substitute the definitions of \cosh(x) and \sinh(x)

Let's substitute \(\cosh x = \frac{e^{x}+e^{-x}}{2}\) and \(\sinh x = \frac{e^{x}-e^{-x}}{2}\) into \((\cosh x)^{2} - (\sinh x)^{2}\) to begin simplifying.

Step 2: Substitute and simplify

The equation becomes \((\frac{e^{x}+e^{-x}}{2})^{2} - (\frac{e^{x}-e^{-x}}{2})^{2}\) which simplifies to \(\frac{e^{2x} + 2 + e^{-2x}}{4} - \frac{e^{2x} - 2 + e^{-2x}}{4}\).

Step 3: Continue Simplification

After combining the two fractions, we get \(\frac{2}{4}\) which simplifies to 0.5. Note that this is a fraction so division is applicable.

Step 4: Finalize the equation

The last step is to convert the decimal to a whole number. This gives the final answer which is 1.

Key Concepts

Hyperbolic Cosine

The hyperbolic cosine function, denoted as \(\cosh x\), is one of the basic hyperbolic functions. It is defined mathematically by \(\cosh x = \frac{e^{x} + e^{-x}}{2}\), where \(e\) represents the base of the natural logarithm.

Think of the hyperbolic cosine as a sort of 'cousin' to the more familiar trigonometric cosine function, but with the distinction that it applies to hyperbolic, rather than circular geometry. Just like the trigonometric functions describe relationships in a circle, hyperbolic functions describe relationships in a hyperbola.

When graphed, \(\cosh x\) has a characteristic smoothed 'V' shape, symmetrical along the y-axis. This symmetry reflects the function's even nature, meaning that \(\cosh(-x) = \cosh(x)\). Understanding this symmetry is key when evaluating hyperbolic identities or solving equations involving \(\cosh x\).

Hyperbolic Sine

In parallel to the hyperbolic cosine, there is the hyperbolic sine function, represented as \(\sinh x\). Its formula is given by \(\sinh x = \frac{e^{x} - e^{-x}}{2}\).

This function might remind you of the regular sine function from trigonometry, but once again, it's tailored to a hyperbolic context. The graph of \(\sinh x\) doesn't repeat as the sine wave does; instead, it continuously increases or decreases in a shape resembling the letter 'S'.

Unlike \(\cosh x\), the hyperbolic sine function is not even; it's odd, which means that \(\sinh(-x) = -\sinh(x)\). This property is extremely useful when proving various hyperbolic identities and when handling transformations of the hyperbolic sine function.

Exponential Functions

Exponential functions form the backbone of both hyperbolic cosine and sine functions. These functions are written in the form \(f(x) = e^{x}\), where \(e\) is the aforementioned base of the natural log.

One of the most remarkable traits of the exponential function is its derivative—\(\frac{d}{dx}e^{x} = e^{x}\)—it's its own derivative!

Moreover, exponentiation has a special property that underlines the behavior of hyperbolic functions. Namely, for any real numbers \(a\) and \(b\), the expressions \(e^{a + b}\) and \(e^{a}e^{b}\) are equivalent. These rules are critical when dealing with the calculus of hyperbolic functions and when proving their various identities, such as \(\cosh^2 x - \sinh^2 x = 1\).

College Algebra

College algebra is the course where many of these concepts come together. Here you learn about functions, their properties, and how they can be manipulated to solve various problems.

Understanding exponential and hyperbolic functions are just a part of the broad spectrum of college algebra. It covers topics from basic equations to the more complex functions, offering a bridge between high school algebra and higher mathematics. College algebra emphasizes the development of problem-solving skills, including proving identities—like the one from our exercise—and solving equations.

The approach used to prove \(\cosh^2 x - \sinh^2 x = 1\) demonstrates the typical logical progression in college algebra: define the functions, manipulate them algebraically, and apply fundamental concepts to arrive at a proof or solution.