Fourier series of exponential term

[8614smith]

Homework Statement

Homework Equations

The Attempt at a Solution

Ive numbered the solution steps, the ones that are giving me trouble are from 1 to 2 and from 3 to 4

From 1 to 2 i don't understand how there can be an exp(x) term taken out of the bracket and still be in the bracket, also don't know how the (1-in) and (1+in) can be taken out but still left in the exp() term

And from 3 to 4 i am assuming its a trig identity but i can't find any on the internet so I am not sure if there's an intermediate step in between these two of whether its a straight swap for an identity

 

[payumooli]

1-2

multiply the nr and dr by complex conjugate of the dr and remove exp(x) which is a common factor

3-4
famous euler's equation
exp(ix) = ...

hope it helped