Chapter A6, Problem 10E

To determine

To simplify:the given equation.

Answer to Problem 10E

The simplified answer is $\text{0}$ .

Explanation of Solution

Given:

  $1\text{n}\left(\text{cosh }x+\text{sinh }x\right)+1\text{n}\left(\text{cosh }x-\text{sinh }x\right).$

Calculation:

It is known that,

  $\text{sinh }x=\frac{e^x-e^{-x}}{2}\text{ cosh }x=\frac{e^x+e^{-x}}{2}$ and ${\text{cosh}}^{2}x-{\text{sinh}}^{2}x=1.$

Therefore,

  $1\text{n}\left(\text{cosh }x+\text{sinh }x\right)+1\text{n}\left(\text{cosh }x-\text{sinh }x\right)=1\text{n}\left(\left(\text{cosh }x+\text{sinh }x\right)\times 1\text{n}\left(\text{cosh }x-\text{sinh }x\right)\right).$

  $=1\text{n}\left({\text{cosh}}^{2}x-{\text{sinh}}^{2}x\right).$

  $\text{=1n 1}\text{.}$

  $\text{=0}\text{.}$

This can also be verified graphically by plotting a graph of

  $y=1\text{n}\left(\text{cosh }x+\text{sinh }x\right)+1\text{n}\left(\text{cosh }x-\text{sinh }x\right).$

Since two equal terms are subtracted from each other which should result in zero. Therefore, If

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

The graph of the function $y=1\text{n}\left(\text{cosh }x+\text{sinh }x\right)+1\text{n}\left(\text{cosh }x-\text{sinh }x\right)$ is shown below.

  

The graph of the function $y=\text{1n }\left(1\right)$ is shown below.

  

The graph of the function $y=1\text{n}\left(\text{cosh }\left(x\right)+\text{sinh }\left(x\right)\right)+1\text{n}\left(\text{cosh }\left(x\right)-\text{sinh }\left(x\right)\right)-\text{1n }\left(1\right)$ is shown below.

  

Conclusion:

The simplified answer is $0$ .