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 \(\operatorname{coth}^{2} x-1=\operatorname{csch}^{2} x\).

Short answer

Question: Verify the identity \(\operatorname{coth}^{2} x-1=\operatorname{csch}^{2} x\). Answer: The given identity is verified as both sides of the equation are equal after using the fundamental identity \(\cosh ^{2} x-\sinh ^{2} x=1\).

Step-by-step solution

Define the involved functions in terms of hyperbolic sine and cosine

We have the following definitions: 1. Hyperbolic cotangent function: \(\operatorname{coth} x = \frac{\cosh x}{\sinh x}\) 2. Hyperbolic cosecant function: \(\operatorname{csch} x = \frac{1}{\sinh x}\)

Rewrite the given identity using the definitions

Using the definitions, we have: \(\operatorname{coth}^{2} x-1=\operatorname{csch}^{2} x\) \(\left(\frac{\cosh x}{\sinh x}\right)^{2} - 1 = \left(\frac{1}{\sinh x}\right)^{2}\)

Simplify the equation

We should now simplify the equation: \(\frac{\cosh^{2} x}{\sinh^{2} x} - 1 = \frac{1}{\sinh^{2} x}\) Now, we can put both terms on the same denominator: \(\frac{\cosh^{2} x - \sinh^{2} x}{\sinh^{2} x} = \frac{1}{\sinh^{2} x}\)

Use the fundamental identity to verify the relationship

We have: \(\frac{\cosh ^{2} x-\sinh ^{2} x}{\sinh^{2} x} = \frac{1}{\sinh^{2} x}\) Using the fundamental identity \(\cosh ^{2} x-\sinh ^{2} x=1\), we get: \(\frac{1}{\sinh^{2} x} = \frac{1}{\sinh^{2} x}\)

Conclusion

Since both sides of the equation are equal, the given identity is verified: \(\operatorname{coth}^{2} x-1=\operatorname{csch}^{2} x\)

Key concepts

Fundamental Hyperbolic Identity

Hyperbolic functions are closely related to exponential functions and have properties similar to trigonometric functions. One of the key concepts in hyperbolic functions is the Fundamental Hyperbolic Identity. This identity is given by: \[\cosh^{2}x - \sinh^{2}x = 1\] This identity mirrors the Pythagorean identity in trigonometry, \(\cos^2\theta + \sin^2\theta = 1\), but with hyperbolic adjustments. Here, \(\cosh x\) and \(\sinh x\) are the hyperbolic counterparts of the cosine and sine functions, respectively. This identity is fundamental because it provides an intrinsic relation between the hyperbolic cosine and the hyperbolic sine. It is the cornerstone for deriving other identities in hyperbolic functions. In the exercise provided, the identity helps to verify the equality between hyperbolic cotangent and cosecant functions' expressions.

Hyperbolic Cotangent

The hyperbolic cotangent function, noted as \(\operatorname{coth} x\), is derived from the hyperbolic cosine and sine functions. It is defined by the ratio: \[\operatorname{coth} x = \frac{\cosh x}{\sinh x}\] This function is analogous to the cotangent function in trigonometry, \(\operatorname{cot} \theta = \frac{\cos \theta}{\sin \theta}\), but it uses hyperbolic functions instead. The cotangent hyperbolic function is typically used in various mathematical and engineering applications where hyperbolic angles or functions appear, such as modeling growth phenomena or waveforms. In the given exercise, \(\operatorname{coth}^2 x\) plays a crucial role in forming and simplifying the identity we aim to verify.

Hyperbolic Cosecant

The hyperbolic cosecant function, denoted as \(\operatorname{csch} x\), complements the sine hyperbolic function. It is an inverse relationship and is described as follows: \[\operatorname{csch} x = \frac{1}{\sinh x}\] Similar to the cosecant function in traditional trigonometry, which is \(\operatorname{csc} \theta = \frac{1}{\sin \theta}\), the hyperbolic cosecant offers an essential tool for simplifying expressions in hyperbolic functions. In the context of the presented exercise, knowing \(\operatorname{csch} x\) enables the transformation and verification of the given identity. It expresses relationships in terms of \(\sinh x\), making it manageable to demonstrate the equality successively.

Verification of Identities

Verification of identities is an essential skill in algebra and calculus involving hyperbolic functions. It involves confirming if a posed equation holds by manipulating and simplifying it using well-known identities. In the exercise, the identity \(\operatorname{coth}^{2} x - 1 = \operatorname{csch}^{2} x\) is a result of dynamic simplification steps. Key steps in verifying this identity include:

• Expressing \(\operatorname{coth} x\) and \(\operatorname{csch} x\) in terms of basic hyperbolic functions.
• Rewriting the expression using these definitions.
• Simplifying the equation until both sides match, using the fundamental hyperbolic identity.

This process involves logical manipulation, showing that both sides are identical under certain transformations, often anchored by fundamental identities, such as \(\cosh^2 x - \sinh^2 x = 1\), ensuring both sides equate accurately.