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A:

The full file is available at Adobe’s website, so you don’t need a crack.
Download and extract the file. Afterwards, open the file with your preferred text editor. If you don’t have a text editor, you can use Notepad.

Q:

Simplify $\frac{1}{\sqrt{x^2-1}}$ in terms of $e^{x}$

I’m trying to simplify $\frac{1}{\sqrt{x^2-1}}$ in terms of $e^{x}$. I already simplified in this way:
$\frac{1}{\sqrt{x^2-1}} = \frac{x}{\sqrt{x^2-1}} = \frac{x}{x\sqrt{1-\frac{1}{x^2}}} = x\sqrt{1-\frac{1}{x^2}} = x – \frac{1}{x}$
Now I need to simplify it in this way:
$x – \frac{1}{x} = e^x – \frac{1}{e^x}$
Can you help me please?

A:

You have the start of a good answer, but have missed the middle. Since $x^2$ is positive we can rewrite the left side as $\sqrt{x^2-1}=\sqrt{(x-\frac 1x)^2}$. Now square both sides and observe that the square root is positive on the right side and positive on the left. You should have
$$
\sqrt{(x-\frac 1x)^2}=e^x-\frac{e^x}{e^x}
$$
$$
\sqrt{(x-\frac 1x)^2}=e^x
$$
since $e^x>\frac 1e$.
You can then simplify this a little:
$$
e^x=\frac{e^x}{e^x}=e^{2x}
$$
Taking common powers is a useful technique for solving exponentials.

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*
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