Why is this step being taken when calculating a fourier series for $e^t$

Question

I am trying to compute the fourier series for $e^t$ in range $-\pi<t<\pi$ and I am stuck at the marked part of the picture.

In the solution it is written that you get $\frac{1+in}{1+n^2}$, why do you have to do it?

Answer 1

That step is just rewriting $\frac{1}{1-in}$ by multiplying top and bottom by $1+in$. This is a routine trick for writing $\frac{a+bi}{c+di}$ as $x+yi$ where $a,b,c,d,x,y$ are all real numbers.

You don't technically need to do this, but it helps when it comes time to add the $n$ term and the $-n$ term to get a representation in terms of sines and cosines instead of complex exponentials.