# Adding exponentials

1. Sep 7, 2013

### leroyjenkens

1. The problem statement, all variables and given/known data

How does $\frac{e^{2iz}+2+e^{-2iz}}{4}=\frac{2}{4}$?

This is part of something more complex, but this should be enough information. I do not see how those exponentials cancel with each other to leave the $\frac{2}{4}$.

Thanks

2. Sep 7, 2013

### HallsofIvy

Staff Emeritus
All I can say is that $(e^{2iz}+ 2+ e^{-2iz})/4$ is NOT equal to "2/4"! For example, if z= 0, the left side is (1+ 2+ 1)/4= 4/4= 1, not 2/4. And since normally you would write "1/2" rather than "2/4", I seems clear to me that you are missing something. What is true is that $e^{2iz}+ 2+ e^{-2iz}= (e^{iz}+ e^{-iz})^2$. Does that help?

Last edited by a moderator: Sep 10, 2013
3. Sep 7, 2013

### leroyjenkens

Thanks. I knew I wasn't that bad at math to not know how to add exponentials.

It's from an example in my book. Here is the whole thing:

Prove that $sin^{2}z+cos^{2}z=1$

$sin^{2}z=(\frac{e^{iz}-e^{-iz}}{2i})^{2}=\frac{e^{2iz}-2+e^{-2iz}}{-4}$

$cos^{2}z=(\frac{e^{iz}+e^{-iz}}{2})^{2}=\frac{e^{2iz}+2+e^{-2iz}}{4}$

$sin^{2}z+cos^{2}z=\frac{2}{4}+\frac{2}{4}=1$

4. Sep 7, 2013

### Saitama

Are you still in a doubt? Check the expansion of $\sin^2z$ in the first step, there's a minus sign in the denominator.

5. Sep 7, 2013

### leroyjenkens

I just tried multiplying the top and bottom of the $sin^{2}z$ expansion by -1 and then adding the two together, I get $\frac{4}{4}$, which is correct. The example makes it look like the exponentials are cancelled before the addition of the two. How did they do that?

Thanks.

6. Sep 7, 2013

### Saitama

What do you get when you do that?

7. Sep 7, 2013

### leroyjenkens

I got $\frac{-e^{2iz}+2-e^{-2iz}}{4}$ and then I added that with the cosine expansion and got 1. But the exponentials didn't disappear until I added the sine and cosine, at which point they cancelled. But in the book, they were gone before the addition of sine and cosine. How did they do that?

Thanks.

8. Sep 7, 2013

### Saitama

They were never gone before the addition. Try to add them yourself.

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