Problem 67 - Calculus Single Variable

Question

Verify the identity $$ \tanh (x+y)=\frac{\tanh (x)+\tanh (y)}{1+\tanh (x) \tanh (y)} $$

Step-by-Step Solution

Answer

The identity is verified: \( \tanh(x + y) = \frac{\tanh(x) + \tanh(y)}{1 + \tanh(x) \tanh(y)} \).

Step 1: Recall the Identity for Hyperbolic Tangent

The hyperbolic tangent of two variables, expressed as \( \tanh(x+y) \), can be expanded using the formula: \[ \tanh(x+y) = \frac{\sinh(x+y)}{\cosh(x+y)} \].

Step 2: Use Sum Formulas for Hyperbolic Functions

Recall the sum formulas for hyperbolic sine and cosine: \( \sinh(x+y) = \sinh(x) \cosh(y) + \cosh(x) \sinh(y) \) and \( \cosh(x+y) = \cosh(x) \cosh(y) + \sinh(x) \sinh(y) \).

Step 3: Substitute Formulas into Hyperbolic Tangent Expression

Substitute the sum formulas into the earlier expression: \[ \tanh(x+y) = \frac{\sinh(x) \cosh(y) + \cosh(x) \sinh(y)}{\cosh(x) \cosh(y) + \sinh(x) \sinh(y)} \].

Step 4: Define Hyperbolic Tangent in Terms of Sinh and Cosh

Recall that \( \tanh(x) = \frac{\sinh(x)}{\cosh(x)} \) and \( \tanh(y) = \frac{\sinh(y)}{\cosh(y)} \). Therefore, \( \sinh(x) = \tanh(x) \cosh(x) \) and \( \sinh(y) = \tanh(y) \cosh(y) \).

Step 5: Substitute Definitions into Tan(x+y) Expression

Substitute \( \sinh(x) \) and \( \sinh(y) \) in terms of \( \tanh(x) \) and \( \tanh(y) \) into the expression: \[ \tanh(x+y) = \frac{\tanh(x) \cosh(x) \cosh(y) + \cosh(x) \tanh(y) \cosh(y)}{\cosh(x) \cosh(y) + \tanh(x) \cosh(x) \tanh(y) \cosh(y)} \].

Step 6: Simplify the Expression

Factor out the common term \( \cosh(x) \cosh(y) \) in the numerator and the denominator: \[ \tanh(x+y) = \frac{\cosh(x) \cosh(y) (\tanh(x) + \tanh(y))}{\cosh(x) \cosh(y) (1 + \tanh(x) \tanh(y))} \].

Step 7: Cancel Common Terms

Cancel the common term \( \cosh(x) \cosh(y) \) from the numerator and the denominator: \[ \tanh(x+y) = \frac{\tanh(x) + \tanh(y)}{1 + \tanh(x) \tanh(y)} \].

Key Concepts

Hyperbolic Tangent Identity

The hyperbolic tangent identity is a crucial part of trigonometry as it extends the familiar tangent function into the hyperbolic realm. The identity is expressed as:

• \( \tanh(x+y) = \frac{\tanh(x) + \tanh(y)}{1 + \tanh(x) \tanh(y)} \)

This identity helps us understand how the hyperbolic tangent of the sum of two variables can be expressed using the hyperbolic tangents of the individual variables.

You can remember it as a parallel to the tangent addition formula used for circular functions. To derive it, we express \(\tanh(x+y)\) as \(\frac{\sinh(x+y)}{\cosh(x+y)}\).

This identity is particularly useful in solving equations involving hyperbolic functions and can greatly simplify complex calculations. It's important to note the symmetry in its form, which makes it especially convenient for mathematical proofs and computations.

Sum Formulas for Hyperbolic Functions

Sum formulas for hyperbolic functions are foundational tools that allow for the simplification and computation of expressions involving hyperbolic sines and cosines. These formulas are similar to those for regular sine and cosine, but instead, they accommodate the properties of hyperbolic curves. Here are the equations:

• \( \sinh(x+y) = \sinh(x) \cosh(y) + \cosh(x) \sinh(y) \)
• \( \cosh(x+y) = \cosh(x) \cosh(y) + \sinh(x) \sinh(y) \)

These sum formulas are essential for expressing the combined effects of two hyperbolic angles. The formula for \(\sinh(x+y)\) captures the additive nature of the hyperbolic sine, while \(\cosh(x+y)\) addresses the essence of the hyperbolic cosine in adding variables.

When used in conjunction with hyperbolic tangent identities, they offer a powerful approach to breaking down complex expressions into manageable forms. This is particularly effective in calculus, where such expressions frequently arise.

Hyperbolic Sine and Cosine

Hyperbolic sine (\(\sinh\)) and cosine (\(\cosh\)) are functions analogous to the regular sine and cosine, but they relate to hyperbolas rather than circles. They are defined through exponential functions as:

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

These functions exhibit interesting properties, such as the identity \(\cosh^2(x) - \sinh^2(x) = 1\), resembling the Pythagorean identity in trigonometry.

Hyperbolic functions have applications across a variety of mathematical fields, including calculus, differential equations, and complex analysis. Their shapes and behaviors mimic some aspects of exponential growth and decay, which makes them valuable in modeling oscillatory and exponential processes.

By understanding \(\sinh\) and \(\cosh\), we gain insight into more abstract mathematical concepts and can tackle problems involving curves that aren't strictly circular, opening the door to diverse scientific applications.