Problem 16 - Calculus Early Transcendentals

Question

Use the given identity to verify the related identity. Use the fundamental identity \(\cosh ^{2} x-\sinh ^{2} x=1\) to verify the identity \(\operatorname{coth}^{2} x-1=\operatorname{csch}^{2} x.\)

Step-by-Step Solution

Answer

Question: Verify the identity $\operatorname{coth}^{2}(x)-1 = \operatorname{csch}^{2}(x)$ using the fundamental identity $\cosh^2(x) - \sinh^2(x) = 1$.

Step 1: Write down the definitions of coth, csch, cosh and sinh

Recall the definitions of the hyperbolic functions: $$\operatorname{coth}(x) = \frac{\cosh(x)}{\sinh(x)},\quad \operatorname{csch}(x) = \frac{1}{\sinh(x)},\quad \cosh(x) = \frac{e^x + e^{-x}}{2}, \quad \sinh(x) = \frac{e^x - e^{-x}}{2}.$$

Step 2: Rewrite the target identity in terms of cosh and sinh

Using the definitions from Step 1, we rewrite the target identity as follows: $$\left(\frac{\cosh(x)}{\sinh(x)}\right)^2 - 1 = \left(\frac{1}{\sinh(x)}\right)^2.$$

Step 3: Simplify the identity by clearing the denominators

To simplify the given identity, first multiply both sides by \(\sinh^2(x)\) to clear the denominators: $$\cosh^2(x)\sinh^2(x) - \sinh^4(x) = 1.$$

Step 4: Use the fundamental identity to simplify the left-hand side

Now, observe that the left-hand side of the equation is $$\cosh^2(x)\sinh^2(x) - \sinh^4(x) = (\sinh^2(x))(\cosh^2(x) - \sinh^2(x)).$$ Recall the fundamental identity $$\cosh^2(x) - \sinh^2(x) = 1.$$ Using this, we can simplify the left-hand side as $$\sinh^2(x)(\cosh^2(x) - \sinh^2(x)) = \sinh^2(x)(1) = \sinh^2(x).$$

Step 5: Establish the target identity

We have now shown that $$\cosh^2(x)\sinh^2(x) - \sinh^4(x) = \sinh^2(x),$$ which is equivalent to the target identity $$\operatorname{coth}^2(x) - 1 = \operatorname{csch}^2(x).$$ Hence, we have successfully verified the target identity.

Key Concepts

Cosh and Sinh

Hyperbolic functions are analogs of trigonometric functions, and two of the most fundamental are the hyperbolic cosine and hyperbolic sine, abbreviated as cosh and sinh, respectively. They are essential for solving a range of problems involving hyperbolic identities.

The hyperbolic cosine function, \[\cosh(x) = \frac{e^x + e^{-x}}{2},\]captures the behavior of exponential growth and decay in a symmetric way. Similarly, the hyperbolic sine function is defined as:\[\sinh(x) = \frac{e^x - e^{-x}}{2},\]showing an unsymmetric exponential behavior that mimics the pattern of sine in trigonometry with different growth for positive and negative directions.

Both functions are vital due to their unique properties:

• They span natural exponential growth and decay in their range of applications.
• They render solutions to certain types of differential equations.
• They help in defining other hyperbolic functions like hyperbolic tangent, secant, and cosecant.

Verifying Identities

Verification of identities involves showing that two expressions are equivalent by transforming one side of the equation to match the other. This process is often essential in mathematics to ensure the validity of equations and functions.

In the context of hyperbolic functions, the task is often to use known identities, such as the fundamental identity:\[\cosh^2(x) - \sinh^2(x) = 1,\]to verify other derived identities. For example, to verify:\[\operatorname{coth}^2(x) - 1 = \operatorname{csch}^2(x),\]we begin by expressing everything in terms of cosh and sinh using fundamental definitions:

• \(\operatorname{coth}(x) = \frac{\cosh(x)}{\sinh(x)}\)
• \(\operatorname{csch}(x) = \frac{1}{\sinh(x)}\)

Simplifying these using substitution and basic algebra often reveals the equivalence, convincing us that the identity holds true under the operations and transformations applied throughout.Verification thus requires a strategic approach where knowing when and how to manipulate expressions crucially assists in validating identities.

Hyperbolic Trigonometric Identities

Hyperbolic trigonometric identities are equations involving hyperbolic functions that hold for all values in their domains. These identities generally mirror their trigonometric counterparts but pertain to the hyperbolic plane.

For instance, the core hyperbolic identity\[\cosh^2(x) - \sinh^2(x) = 1,\]is analogous to the Pythagorean identity \ \(\sin^2\theta + \cos^2\theta = 1\/\ \) found in classic trigonometry, showing symmetry in structure but splitting under different variances.

Other significant identities include relational definitions of hyperbolic functions in terms of cosh and sinh:

• \(\operatorname{tanh}(x) = \frac{\sinh(x)}{\cosh(x)}\)
• \(\operatorname{sech}(x) = \frac{1}{\cosh(x)}\)
• \(\operatorname{csch}(x) = \frac{1}{\sinh(x)}\)
• \(\operatorname{coth}(x) = \frac{\cosh(x)}{\sinh(x)}\)

Understanding and using these identities facilitates solving complex mathematical problems, particularly in calculus and differential equations, as they provide tools for simplification and substitution in broader contexts.