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Chapter 5: Problem 16

Prove the identity. \(\tanh (x+y)=\frac{\tanh x+\tanh y}{1+\tanh x \tanh y}\)

Short Answer

Expert verified
To prove the identity $\tanh(x+y) = \frac{\tanh x + \tanh y}{1 + \tanh x \tanh y}$, start by writing the definitions of hyperbolic functions and expanding $\tanh(x+y)$ using the addition formulas for $\sinh(x+y)$ and $\cosh(x+y)$. Then substitute the definitions of $\tanh x$ and $\tanh y$ in the expression, simplify it, and finally show that it indeed equals the given identity. The proof is: \[ \tanh(x+y) = \frac{\sinh(x+y)}{\cosh(x+y)} = \frac{\sinh(x)\cosh(y)+\cosh(x)\sinh(y)}{\cosh(x)\cosh(y)+\sinh(x)\sinh(y)} = \frac{\tanh x + \tanh y}{1 +\tanh x \tanh y} \]

Step by step solution

01

Write the definitions of hyperbolic functions

Recall that the hyperbolic tangent can be defined in terms of hyperbolic sine and hyperbolic cosine functions, as: \[ \tanh(x)=\frac{\sinh(x)}{\cosh(x)} \] Also, remember the addition formulas for \(\sinh(x+y)\) and \(\cosh(x+y)\): \[ \sinh(x+y)=\sinh(x)\cosh(y)+\cosh(x)\sinh(y) \] \[ \cosh(x+y)=\cosh(x)\cosh(y)+\sinh(x)\sinh(y) \]
02

Expand \(\tanh(x+y)\) using the addition formulas

Using the definition of \(\tanh(x)\) and addition formulas for \(\sinh(x+y)\) and \(\cosh(x+y)\), we can expand \(\tanh(x+y)\) as follows: \[ \tanh(x+y)=\frac{\sinh(x+y)}{\cosh(x+y)}=\frac{\sinh(x)\cosh(y)+\cosh(x)\sinh(y)}{\cosh(x)\cosh(y)+\sinh(x)\sinh(y)} \]
03

Substitute the definitions of \(\tanh x\) and \(\tanh y\)

Next, we substitute the definitions of \(\tanh x\) and \(\tanh y\) in the expanded expression of \(\tanh(x+y)\): \[ \frac{\sinh(x)\cosh(y)+\cosh(x)\sinh(y)}{\cosh(x)\cosh(y)+\sinh(x)\sinh(y)} =\frac{\cosh(y)\frac{\sinh(x)}{\cosh(x)}+\cosh(x)\frac{\sinh(y)}{\cosh(y)}}{\cosh(x)\cosh(y)+\sinh(x)\sinh(y)} \]
04

Simplify the expression

Now, let's simplify the expression by multiplying the numerators and denominators by \(\cosh(x)\cosh(y)\): \[ \frac{\cosh(y)\frac{\sinh(x)}{\cosh(x)}+\cosh(x)\frac{\sinh(y)}{\cosh(y)}}{\cosh(x)\cosh(y)+\sinh(x)\sinh(y)} = \frac{(\sinh(x)\cosh(x)\cosh(y)+\cosh(x)\cosh(y)\sinh(y)}{\cosh^2(x)\cosh(y)+\sinh(x)\sinh(y)\cosh(y)} \] Next, divide both numerator and denominator by \(\cosh(x) \cosh(y)\): \[ \frac{(\sinh(x)\cosh(x)\cosh(y)+\cosh(x)\cosh(y)\sinh(y)}{\cosh^2(x)\cosh(y)+\sinh(x)\sinh(y)\cosh(y)} = \frac{\sinh(x) + \sinh(y)}{\cosh(x)\cosh(y) + \sinh(x)\sinh(y)} \]
05

Write the final result using definitions of hyperbolic functions

To express the simplified expression in terms of \(\tanh x\) and \(\tanh y\), substitute the definitions of \(\tanh x\) and \(\tanh y\) back into the expression: \[ \frac{\sinh(x) + \sinh(y)}{\cosh(x)\cosh(y) + \sinh(x)\sinh(y)} = \frac{\tanh x + \tanh y}{1 +\tanh x \tanh y} \] This completes the proof: \[ \tanh(x+y) = \frac{\tanh x + \tanh y}{1 + \tanh x \tanh y} \]

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Key Concepts

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

Hyperbolic Functions
Hyperbolic functions, including tanh (hyperbolic tangent), sinh (hyperbolic sine), and cosh (hyperbolic cosine), are analogs of the trigonometric functions, but for a hyperbola instead of a circle. They have important applications in many fields such as calculus, physics, and engineering.

Hyperbolic functions are defined using the exponential function e. For instance, sinh x is given by \[\begin{equation}\sinh(x) = \frac{e^x - e^{-x}}{2}\end{equation}\]while cosh x is defined as\[\begin{equation}\cosh(x) = \frac{e^x + e^{-x}}{2}\end{equation}\]As such, tanh x, or the hyperbolic tangent of x, is the ratio of the hyperbolic sine to the hyperbolic cosine:\[\begin{equation}\tanh(x)=\frac{\sinh(x)}{\cosh(x)}\end{equation}\]These functions have properties that mirror the familiar sine and cosine functions, but with different graphical representations and identities. Understanding these properties is crucial for solving problems involving hyperbolic functions.
Proof of Hyperbolic Identities
Proving hyperbolic identities, such as the tanh addition formula, involves a process similar to proving trigonometric identities. It often requires substituting definitions, using known identities, and algebraic manipulations.

The proof starts by expressing the hyperbolic functions involved in terms of exponential functions. By using the addition formulas for sinh and cosh, one can proceed to expand terms and use substitution to express everything in terms of tanh. After simplifying, the original expression reveals the underlying identity. These steps showcase the importance of having a solid understanding of not just hyperbolic functions but also their interrelations and how they can be manipulated algebraically.

For example, proving the addition formula for tanh requires recognizing patterns, rearranging terms, and simplifying fractions, as can be seen in the step-by-step solution provided. Such proofs not only test one's knowledge of hyperbolic functions but also enhance algebraic skills and the ability to handle complex mathematical expressions.
Simplifying Hyperbolic Expressions
Simplifying hyperbolic expressions is a valuable skill that allows for easier computation and better understanding of the properties of hyperbolic functions. Like algebraic expressions, hyperbolic expressions can be simplified by combining like terms, factoring, and using hyperbolic identities.

To simplify a hyperbolic expression, one may need to multiply or divide by certain terms to clear complex denominators or reduce fractions, as seen in the tanh addition formula proof. Careful manipulation often leads to a form where an identity can be applied, simplifying the expression further.

As students work through hyperbolic function exercises, such as the tanh addition formula, they gain practice and develop the ability to recognize when and how to simplify expressions effectively. This not only aids in solving mathematical and real-world problems but also in preparing for more advanced topics in mathematics.

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Most popular questions from this chapter

Find the centroid of the region bounded by the graphs of the given equations. $$ y=-x^{2}+3, \quad y=x^{2}-2 x-1 $$

(a) plot the graph of the function \(f\). (b) write an integral giving the arc length of the graph of the function over the indicated interval, and (c) find the arc length of the curve accurate to four decimal places. $$ f(x)=\frac{x}{1+x^{4}} ; \quad[0,1] $$

Heat-Seeking Missiles In a test conducted on a heat-seeking Missile \(A\), the target missile \(B\), which is initially at a distance of \(b\) miles from Missile \(A\), is launched vertically upward. Assume that Missile \(A\) travels at a constant speed \(v_{A}\), that Missile \(B\) travels at a constant speed \(v_{B}\left(v_{A}>v_{B}\right)\), and that Missile \(A\), which is launched from the origin, is always pointed at Missile \(B\). Then the trajectory of Missile \(A\) is $$ y=\frac{b}{2}\left[\frac{\left(1-\frac{x}{b}\right)^{1+c}}{1+c}-\frac{\left(1-\frac{x}{b}\right)^{1-c}}{1-c}\right]+\frac{b c}{1-c^{2}} $$ where \(c=v_{B} / v_{A}\). The trajectory of Missile \(A\) is a pursuit curve. a. Find the point at which Missile \(A\) intercepts Missile \(B\). b. Show that $$ \frac{d y}{d x}=-\sinh \left[c \ln \left(1-\frac{x}{b}\right)\right] $$ c. Suppose that \(b=1\) and \(c=\frac{1}{2}\). Show that the distance \(D\) traveled by Missile \(A\) for the intercept is \(1 \frac{1}{3} \mathrm{mi}\). Hint: \(D=\int_{0}^{1} \sqrt{1+\left(\frac{d y}{d x}\right)^{2}} d x\) d. Plot the graph of the trajectory of the heat-seeking missile taking \(b=1\) and \(c=\frac{1}{2}\)

Use the method of disks or washers, or the method of cylindrical shells to find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the indicated axis. Sketch the region and a representative rectangle. \(y=(x-1)^{2}, \quad y=x+1 ;\) the \(x\) -axis

In Exercises 31 and 32, use differentials to approximate the arc length of the graph of the equation from \(P\) to \(Q\). $$ y=x^{3}+1 ; \quad P(1,2), Q(1.2,2.728) $$
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