What are the missing steps in this Fourier Series problem

Question

Problem here

Please help me figure out the steps in between the two expressions separated by the red arrow.

I am aware of the identity $$ e^ {-j0.5\pi k} = \cos(0.5\pi k)-j\sin(0.5\pi k) $$ but I am not sure how to proceed.

Answer 1

$$(\delta[k-1]+\delta[k+1])e^{-j0.5\pi k}=\delta[k-1]e^{-j0.5\pi k}+\delta[k+1]e^{-j0.5\pi k}$$ Now note that $f(x)\delta(x-x_0)=f(x_0)\delta(x-x_0)$. Why is that you might ask. If $x\ne x_0$ the delta function is $0$, so both sides of the equation are $0$. Now you go back to the first equation, and you can write $$\delta[k-1]e^{-j0.5\pi k}+\delta[k+1]e^{-j0.5\pi k}=\delta[k-1]e^{-j0.5\pi 1}+\delta[k+1]e^{-j0.5\pi (-1)}$$ You have $$e^{j 0.5\pi}=\cos(0.5\pi)+j\sin(0.5\pi)=j\\e^{-j 0.5\pi}=\cos(-0.5\pi)+j\sin(-0.5\pi)=-j$$

So then $$(\delta[k-1]+\delta[k+1])e^{-j0.5\pi k}=-j\delta[k-1]+j\delta[k+1]$$