1. ## formula rearranged

Just wondering if it was possible for someone to show me how you get from this equation:

ie^(it)/e^(-it)-1

to this

i/1-e^(-it)
Cheers in advance! Its much appreciated.

2. If you mean: is $\displaystyle \frac{ie^{it}}{e^{-it}-1}=\frac{i}{1-e^{-it}}$, the answer is no.

Suppose $\displaystyle \frac{ie^{it}}{e^{-it}-1}=\frac{i}{1-e^{-it}}$. Then, by cancelling $\displaystyle i$ we get $\displaystyle \frac{e^{it}}{e^{-it}-1}=\frac{1}{1-e^{-it}}$. By multiplying the LHS by $\displaystyle \frac{-1}{-1}$, we get $\displaystyle \frac{-e^{it}}{1-e^{-it}}=\frac{1}{1-e^{-it}}$. Finally we end up with $\displaystyle -e^{it}=1$, which is a contradiction.

3. Originally Posted by Karty
Just wondering if it was possible for someone to show me how you get from this equation:

ie^(it)/e^(-it)-1

to this

i/1-e^(-it)
Cheers in advance! Its much appreciated.
hi Karty,

you are losing a factor...

$\displaystyle \frac{ie^{it}}{e^{-it}-1}=\frac{-ie^{it}}{-(e^{-it}-1)}$

$\displaystyle =\frac{-ie^{it}}{1-e^{-it}}=-e^{it}\ \frac{i}{1-e^{-it}}$

Either that, or you have a typo,

$\displaystyle \frac{ie^{-it}}{e^{-it}-1}=\frac{ie^{-it}e^{it}}{e^{it}\left(e^{-it}-1\right)}=\frac{i}{1-e^{it}}$

or

$\displaystyle \frac{ie^{it}}{e^{it}-1}=\frac{ie^{it}e^{-it}}{e^{-it}\left(e^{it}-1\right)}$

which leads to the result you've given

4. Originally Posted by Karty
ie^(it)/e^(-it)-1
As you show it:
ie^(it)/e^(-it) - 1
= ie^(it)e^(it) - 1
= ie^(2it) - 1

5. my mistake, there was an error in the formula:

ie^(-it)/e^(-it)-1

to this

i/1-e^(-it)

sorry to confuse, and thanks for taking the time to answer my qs!

Karty