Chapter A6, Problem 8E

To determine

To simplify:the given equation.

Answer to Problem 8E

The simplified answer is $e^{-3x}$ .

Explanation of Solution

Given:

  $\text{cosh 3}x-\text{sinh 3}x.$

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,

  $\begin{array}{l}\text{cosh 3}x-\text{sinh 3}x=\frac{e^{3x}+e^{-3x}}{2}-\frac{e^{3x}-e^{-3x}}{2}.\\ =e^{-3x}.\end{array}$

This can also be verified graphically by plotting a graph of

  $y=\text{cosh 3}x-\text{sinh 3}x-e^{-3x}.$

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

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

The graph of the function $y=\text{cosh 3}x-\text{sinh 3}x$ is shown below.

  

The graph of the function $y=e^{-3x}$ is shown below.

  

The graph of the function $y=\text{cosh 3}x-\text{sinh 3}x-e^{-3x}$ is shown below.

  

Conclusion:

The simplified answer is $e^{-3x}$ .