Prove questions similar to the binomial theorem for complex numbers

Question

1

How should I prove that $$\forall n \in \mathbb{Z}, n\geq0, (1+i)^{n}+(1-i)^{n}=2\sum_{k=0}^{\left\lfloor \frac{n}{2}\right\rfloor}\binom{n}{2k}(-1)^{k}$$

From what I can see here, it looks really similar to the binomial theorem, but alternates between negative and positive numbers. Is it possible for me to invoke De Moivre's Theorem in this case?

2

Also, a similar question that I wanted to prove was $$\forall n \in \mathbb{Z}, n\geq0, (1+i)^{n}+(1-i)^{n}=0\leftrightarrow n\equiv2(mod \text{ 4})$$

This seems to be a very tricky if and only if statement, but I made a very key observation that $(1+i)^{4}=(1-i)^{4}=-4$. I'm a bit lost as to how I can apply it though. Especially to both the forward and backward implications.

Answer 1

Hint for 1:

Expand the binomials $(1+i)^n$ and $(1-i)^n$ and add them, noting that $i^2=(-i)^2=-1$.

Hint for 2:

It does help here to use de Moivre's theorem,

because $1+i=\sqrt2e^{i \pi/4}$ and $1+i=\sqrt2e^{- i \pi/4}$,

so $(1+i)^n+(1-i)^n=\sqrt2^n(e^{in\pi/4}+e^{-in\pi/4})=\sqrt2^n2\cos(n\pi/4).$