Problem 10 - Essential Calculus Early Transcendentals

Question

Prove the identity. $$ \cosh x-\sinh x=e^{-x} $$

Step-by-Step Solution

Answer

The identity is proven: \(\cosh x - \sinh x = e^{-x}\).

Step 1: Use Definition of Hyperbolic Functions

Recall the definitions of the hyperbolic cosine and hyperbolic sine functions: \[\cosh x = \frac{e^x + e^{-x}}{2} \quad \text{and} \quad \sinh x = \frac{e^x - e^{-x}}{2}\]

Step 2: Write the Given Expression

Using the definitions from Step 1, we can write the expression \(\cosh x - \sinh x\) as follows:\[\cosh x - \sinh x = \left(\frac{e^x + e^{-x}}{2}\right) - \left(\frac{e^x - e^{-x}}{2}\right)\]

Step 3: Simplify the Expression

Combine the fractions by subtracting the two expressions inside the parentheses:\[\cosh x - \sinh x = \frac{e^x + e^{-x} - e^x + e^{-x}}{2}\]The \(e^x\) terms will cancel each other out.

Step 4: Perform Final Simplification

Continue from Step 3 to further simplify:\[\frac{2e^{-x}}{2} = e^{-x}\]So, \(\cosh x - \sinh x = e^{-x}\).

Step 5: Conclusion

We have shown that \(\cosh x - \sinh x = e^{-x}\), thus proving the identity is correct.

Key Concepts

cosh

The hyperbolic cosine function, denoted as \(\cosh x\), is a fundamental component of hyperbolic functions, which are similar to the usual trigonometric functions but are based on exponential functions. The definition for \(\cosh x\) is:

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

This definition highlights the intrinsic connection between \(\cosh x\) and exponential functions. By taking the average of \(e^x\) and \(e^{-x}\), \(\cosh x\) exhibits symmetry similar to the cosine function, yet it extends over the hyperbolic plane.
\(\cosh x\) is always positive, and it resembles the shape of a half-parabola that opens upwards when graphed. This property is valuable when analyzing hyperbolic angles and areas.

sinh

The hyperbolic sine function, denoted as \(\sinh x\), is a companion to \(\cosh x\) and operates within the set of hyperbolic functions to model different hyperbolic phenomena. Its expression is:

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

Distinct from \(\cosh x\), the function \(\sinh x\) calculates the difference between \(e^x\) and \(e^{-x}\), then divides by two. This involvement of both exponential increases gives \(\sinh x\) its true origin as a hyperbolic analog to the sine function.
\(\sinh x\) provides values that are symmetric about the origin in nature; it can be both positive and negative. It appears as an S-shaped curve, accounting for the positive values when \(x\) is positive and negative values when \(x\) is negative. This symmetry around the y-axis allows it to be very useful in mathematical applications including solving hyperbolic differential equations.

identity proof

An identity proof is a methodical process of demonstrating that two expressions are equivalent across all possible values of their variables. In this context, we seek to prove the identity \(\cosh x - \sinh x = e^{-x}\). This proof relies on substituting the expressions of \(\cosh x\) and \(\sinh x\) into the left-hand side of the equation.
Using the definitions:

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

We substitute and simplify:

• \(\cosh x - \sinh x = \left(\frac{e^x + e^{-x}}{2}\right) - \left(\frac{e^x - e^{-x}}{2}\right)\)
• Combine and simplify terms inside the fraction: \(\frac{e^x + e^{-x} - e^x + e^{-x}}{2} = \frac{2e^{-x}}{2}\)
• This reduces to just \(e^{-x}\), completing the proof.

The effectiveness of such proofs lies in their logical progression and adherence to mathematical rules.

exponential functions

Exponential functions, in their core form \(f(x) = e^x\), serve as pivotal elements in understanding hyperbolic functions. They are characterized by a constant base \(e\) (approximately 2.71828), raised to a variable exponent.
Exponential functions are key due to their unique rate of growth and decay:

• When \(x\) is positive \(e^x\) grows indefinitely, while \(e^{-x}\), representing decay, decreases towards zero.
• Exponential decay, exemplified by \(e^{-x}\), is fundamental in many natural processes like radioactive decay and population dynamics.

In the hyperbolic context, the functions \(\cosh\) and \(\sinh\) utilize \(e^x\) and \(e^{-x}\) to effortlessly mix growth and decay patterns. This compatibility creates a bridge between exponential behaviors and the geometry of the hyperbolic plane, vital for solving hyperbolic identities and exploring their applications in physics and engineering.