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Chapter 7: Problem 16

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

Short Answer

Expert verified
Question: Prove the identity \(\coth^2 x - 1 = \operatorname{csch}^2 x\) using the fundamental hyperbolic identity \(\cosh^2 x - \sinh^2 x = 1\). Solution: We follow these steps: 1. Define coth and csch functions: \(\coth x = \frac{\cosh x}{\sinh x}\) and \(\operatorname{csch} x = \frac{1}{\sinh x}\). 2. Express coth^2 x in terms of cosh and sinh functions: \(\coth^2 x = \frac{\cosh^2 x}{\sinh^2 x}\). 3. Express csch^2 x in terms of cosh and sinh functions: \(\operatorname{csch}^2 x = \frac{1}{\sinh^2 x}\). 4. Show that \(\coth^2 x - 1 = \operatorname{csch}^2 x\): \( \coth^2 x - 1 = \frac{\cosh^2 x - \sinh^2 x}{\sinh^2 x} = \frac{1}{\sinh^2 x} = \operatorname{csch}^2 x \).

Step by step solution

01

Define coth and csch functions

Recall the definitions of the hyperbolic cotangent function (coth) and the hyperbolic cosecant function (csch): \(\coth x = \frac{\cosh x}{\sinh x}\) and \(\operatorname{csch} x = \frac{1}{\sinh x}\).
02

Express coth^2 x in terms of cosh and sinh functions

Using the definition of coth, we can express \(\coth^2 x\) as follows: \(\coth^2 x = \left(\frac{\cosh x}{\sinh x}\right)^2 = \frac{\cosh^2 x}{\sinh^2 x}\).
03

Express csch^2 x in terms of cosh and sinh functions

Using the definition of csch, we can express \(\operatorname{csch}^2 x\) as follows: \(\operatorname{csch}^2 x = \left(\frac{1}{\sinh x}\right)^2 = \frac{1}{\sinh^2 x}\).
04

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

Now, we need to show that \(\coth^2 x - 1 = \operatorname{csch}^2 x\). We can do this by using the expressions for \(\coth^2 x\) and \(\operatorname{csch}^2 x\) that we found in Steps 2 and 3: \(\coth^2 x - 1 = \frac{\cosh^2 x}{\sinh^2 x} - 1 = \frac{\cosh^2 x - \sinh^2 x}{\sinh^2 x}\). From the fundamental identity given in the exercise, we know that \(\cosh^2 x - \sinh^2 x = 1\). Therefore, we can substitute this into the equation above: \(\frac{\cosh^2 x - \sinh^2 x}{\sinh^2 x} = \frac{1}{\sinh^2 x}\). This expression is identical to the expression for \(\operatorname{csch}^2 x\) that we found in Step 3: \(\frac{1}{\sinh^2 x} = \operatorname{csch}^2 x\). Thus, we have proven the given identity: \(\coth^2 x - 1 = \operatorname{csch}^2 x\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

coth
The hyperbolic cotangent function, denoted as coth, is analogous to the regular trigonometric cotangent function but applied to hyperbolic angles. The defining formula for the hyperbolic cotangent is: \( \coth x = \frac{\cosh x}{\sinh x} \). This function plays a crucial role in expressing relationships within hyperbolic identities.

To understand \( \coth x \) better, it's important to note the behavior of the hyperbolic functions \( \cosh x \) and \( \sinh x \):
  • \( \cosh x = \frac{e^x + e^{-x}}{2} \) which represents how an exponential function behaves symmetrically around zero.
  • \( \sinh x = \frac{e^x - e^{-x}}{2} \) which captures the asymmetric properties between positive and negative exponents.
This relationship allows you to use \( \coth x \) effectively in simplifying and proving identities, as demonstrated in proving that \( \coth^2 x - 1 = \operatorname{csch}^2 x \).
csch
The hyperbolic cosecant function, symbolized as csch, relates similarly to its trigonometric counterpart, the cosecant function, but in the hyperbolic domain. It is defined by the formula: \( \operatorname{csch} x = \frac{1}{\sinh x} \). This function is vital in hyperbolic trigonometry for expressing inverses of hyperbolic sine relationships.

Examining \( \operatorname{csch} x \) involves understanding its inverse nature to \( \sinh x \):
  • The hyperbolic sine \( \sinh x \) increases exponentially, meaning \( \operatorname{csch} x \) will decrease exponentially with increasing \( x \).
  • \( \operatorname{csch} x \) becomes undefined at \( x = 0 \) since it involves division by zero, much like its trigonometric equivalent.
In hyperbolic identities, \( \operatorname{csch}^2 x \) clearly demonstrates that it can be derived using existing hyperbolic identities, such as the one proven \( \coth^2 x - 1 = \operatorname{csch}^2 x \).
cosh and sinh identities
Hyperbolic functions have identities akin to their trigonometric counterparts. One fundamental identity is \( \cosh^2 x - \sinh^2 x = 1 \), reminiscent of the Pythagorean identity. This provides a critical foundation for proving many equations involving hyperbolic functions.

Key aspects of \( \cosh x \) and \( \sinh x \) include:

  • \( \cosh x \) reflects around the y-axis, always giving positive results, a parallel to the cos function.
  • \( \sinh x \) crosses the origin, providing both positive and negative outputs based on \( x \), akin to the sin function.
By using their behavior and identities, you can simplify and transform complex expressions. For example, using \( \cosh^2 x = \sinh^2 x + 1 \), you can rearrange and substitute to prove relationships like \( \coth^2 x - 1 = \operatorname{csch}^2 x \) by presenting equivalent forms in \( \sinh \) and \( \cosh \).
proof technique
Proving hyperbolic identities often involves expressing functions in terms of their fundamental definitions and substituting known identities. This approach simplifies complex hyperbolic expressions.

A typical proof process starts by:
  • Recalling and using definitions: Convert compound expressions into simpler ones using basic hyperbolic functions.
  • Leveraging known identities: Utilize identities like \( \cosh^2 x - \sinh^2 x = 1 \) to make substitutions.
  • Simplifying arithmetic: Carefully manipulate algebraic expressions through substitution to achieve the desired form.
In the case of proving \( \coth^2 x - 1 = \operatorname{csch}^2 x \), this method entails substituting the definitions \( \coth^2 x = \frac{\cosh^2 x}{\sinh^2 x} \) and \( \operatorname{csch}^2 x = \frac{1}{\sinh^2 x} \), and leveraging the fundamental identity \( \cosh^2 x - \sinh^2 x = 1 \) to simplify and complete the proof thoroughly.

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