Chapter A6, Problem 9E

To determine

To simplify:the given equation.

Answer to Problem 9E

The simplified answer is $e^{4x}$ .

Explanation of Solution

Given:

  ${\left(\text{sinh }x+\text{cosh }x\right)}^4.$

Calculation:

It is known that

  $\text{sinh }x=\frac{e^x-e^{-x}}{2}$ and $\text{cosh }x=\frac{e^x+e^{-x}}{2}.$

Therefore,

  $\begin{array}{l}{\left(\text{sinh }x+\text{cosh }x\right)}^4={\left(\frac{e^x+e^{-x}}{2}+\frac{e^x-e^{-x}}{2}\right)}^4.\\ ={\left(e^x\right)}^4.\\ {\text{ =e}}^{4x}.\end{array}$

This can also be verified graphically by plotting a graph of

  $y={\left(\text{sinh }x+\cosh x\right)}^4-e^{4x}.$

Sinceequal terms are subtracted from each other whichshould result in zero. Therefore, If

This graph comes similar to $y=0$ , then our solution is correct.

The graph of the function ${\left(\text{sinh }x+\text{cosh }x\right)}^4$ is shown below.

  

The graph of the function $y=e^{4x}$ is shown below.

  

The graph of the function $y={\left(\text{sinh }x+\text{cosh }x\right)}^4-e^{4x}$ is shown below.

  

Conclusion:

The simplified answer is $e^{4x}$ .