Intuition about simplifying this expression

[Ikari]

Homework Statement

I was solving a problem and came to a part where I was dealing with the expression,

1 - e^-t
--------
1 - e^t

It turns out that this can be simplified to -e^-t but I had no idea from simply looking at the first expression that it could be simplified, and I tried to continue using a cludgy quotient rule to get the derivative, where if I had have simplified it, it would have been much easier. Is this simply a matter of being inexperienced, or am I missing some concept in elementary algebra that would let me recognize a potential simplification? For instance, I can easily recognize difference of squares, etc, but not something like this. How can I become a stronger mathematician moving forward in this regard?

Thanks in advance!

Homework Equations

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The Attempt at a Solution

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[tnich]

Homework Statement

I was solving a problem and came to a part where I was dealing with the expression,

1 - e^-t
--------
1 - e^t

It turns out that this can be simplified to -e^-t but I had no idea from simply looking at the first expression that it could be simplified, and I tried to continue using a cludgy quotient rule to get the derivative, where if I had have simplified it, it would have been much easier. Is this simply a matter of being inexperienced, or am I missing some concept in elementary algebra that would let me recognize a potential simplification? For instance, I can easily recognize difference of squares, etc, but not something like this. How can I become a stronger mathematician moving forward in this regard?

Thanks in advance!

Homework Equations

N/A

The Attempt at a Solution

N/A[/B]

Don't beat yourself up about this. We all have cases where the answer seems obvious after we see what it is.

 

[Ray Vickson]

Homework Statement

I was solving a problem and came to a part where I was dealing with the expression,

1 - e^-t
--------
1 - e^t

It turns out that this can be simplified to -e^-t but I had no idea from simply looking at the first expression that it could be simplified, and I tried to continue using a cludgy quotient rule to get the derivative, where if I had have simplified it, it would have been much easier. Is this simply a matter of being inexperienced, or am I missing some concept in elementary algebra that would let me recognize a potential simplification? For instance, I can easily recognize difference of squares, etc, but not something like this. How can I become a stronger mathematician moving forward in this regard?

Thanks in advance!

Homework Equations

N/A

The Attempt at a Solution

N/A[/B]

Sometimes it helps to change the symbols: let $y = e^t$. Then your fraction is
$$ \text{fraction} = \frac{1 - \frac 1 y}{1-y} = \frac{ \frac{y-1}{y}}{1-y}.$$
Can you see now how the simplification would proceed?

Anyway, after simplifying, put back $e^t$ in place of $y,$ (or $e^{-t}$ in place of $1/y$).