Problem 8 - Calculus of a Single Variable

Question

In Exercises 7–14, verify the identity. $$ \operatorname{coth}^{2} x-\operatorname{csch}^{2} x=1 $$

Step-by-Step Solution

Answer

The identity is verified by expressing \( \operatorname{coth} x\) and \(\operatorname{csch} x\) in terms of hyperbolic sine and cosine and simplifying the equation, concluding that the identity is true.

Step 1: Express hyperbolic cotangent in terms of hyperbolic cosine and sine

According to the formula, hyperbolic cotangent is given by \( \operatorname{coth} x = \frac {\operatorname{cosh} x} {\operatorname{sinh} x}\), we can substitute this to the equation to obtain \( \left( \frac {\operatorname{cosh} x} {\operatorname{sinh} x} \right) ^2 - \operatorname{csch}^{2} x = 1\).

Step 2: Express hyperbolic cosecant in terms of hyperbolic sine

We know that the hyperbolic cosecant is the reciprocal of the hyperbolic sine, therefore \( \operatorname{csch} x = \frac {1} {\operatorname{sinh} x}\), replacing csch in the equation yields : \( \left( \frac {\operatorname{cosh} x} {\operatorname{sinh} x} \right) ^2 - \left( \frac {1} {\operatorname{sinh} x} \right) ^2 = 1\).

Step 3: Simplify the Equation

The equation now becomes \( \frac {\operatorname{cosh}^{2} x - 1} {\operatorname{sinh}^{2} x} = 1\). Given that \(\operatorname{cosh}^{2} x - 1 = \operatorname{sinh}^{2} x\), this simplifies to \( \frac {\operatorname{sinh}^{2} x} {\operatorname{sinh}^{2} x} = 1 \) simplifying further gives 1 = 1

Step 4: Conclusion

Therefore, the identity \( \operatorname{coth}^{2} x - \operatorname{csch}^{2} x = 1\) is verified.

Key Concepts

Hyperbolic Cotangent

The hyperbolic cotangent is an essential function in hyperbolic trigonometry. It is defined similarly to the classical trigonometric functions but uses hyperbolic functions. The hyperbolic cotangent of an angle \( x \) is given by the ratio of the hyperbolic cosine to the hyperbolic sine. Specifically, it is expressed as:

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

This definition showcases it as a hyperbolic analog to the cotangent found in ordinary trigonometry. In most problems, understanding how to express hyperbolic cotangent in terms of other hyperbolic functions provides a pathway to solve more complex expressions and verify identities. It is crucial to be comfortable with transitioning between these forms for ease of computation.

Hyperbolic Cosecant

The hyperbolic cosecant function supplements the suite of hyperbolic functions much like cosecant does in regular trigonometry. It is defined as the reciprocal of the hyperbolic sine. Specifically:

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

This expression is particularly useful when simplifying equations in hyperbolic function contexts, for instance, while verifying an identity. Understanding hyperbolic cosecant assists in breaking down complex equations because it allows for the expression of terms in simpler reciprocal forms, which can often make a problem easier to solve, like the identity verification provided in this exercise.

Verifying Identities

Verifying mathematical identities is about proving that two expressions are equivalent for all values in their domains. In the realm of hyperbolic functions, identities can be verified by using known definitions and properties of these functions.
For example, verifying the identity \( \operatorname{coth}^2 x - \operatorname{csch}^2 x = 1 \) involves expressing each hyperbolic function in its fundamental form and simplifying the expression to the point where both sides of the equation show the same value.

• Break down complex expressions based on known hyperbolic trigonometric identities and substitutions.
• Systematically simplify and manipulate expressions using algebraic methods.
• Conclude the verification once the identity holds true for the expressions calculated.

By systematically working through these steps, as showcased in the provided solution, verifying identities becomes a streamlined process.

Simplifying Trigonometric Expressions

Simplifying trigonometric expressions, especially those involving hyperbolic functions, relies heavily on familiarity with fundamental identities and properties of hyperbolic functions. The goal is often to convert an expression into a simpler or more useful form.
Here are key approaches to simplifying expressions:

• Use definitions: Convert complex terms into basic hyperbolic functions using their definitions or reciprocal identities.
• Apply identities: Use hyperbolic identities such as \( \operatorname{cosh}^2 x - \operatorname{sinh}^2 x = 1 \) to substitute or reduce terms.
• Solve for desired form: Simplify further until the equation or expression reaches the simplest form or the desired result.

Understanding and applying these methods is crucial to mastering the simplification of hyperbolic trigonometric expressions and can be directly applied to solve diverse kinds of problems efficiently.