$\cos(2\pi f nT +2\pi N_n) =\ cos(2\pi f nT)$ and more, why?

Question

I am studying signals and systems and this came up?

Could someone explain why is $\cos(2\pi f nT +2\pi N_n)$ equal to $\ cos(2\pi f nT)$?

The book says: "because $N_n$ is an integer"

I am wondering how does that explain the equation?

Same with the following:

$e^{i2\pi Nf_snT}=1$

in

snippet from book

Answer 1

Look at the picture below;

Can you see that between $0$ and $2\pi$ has a specific behaviour, but repeats after the $2\pi$? This is because the cosine function has a period of $2\pi$. That is to say, $$\cos(x) = \cos(x + 2k\pi)$$

for $k\in \Bbb{Z}.$ To illustrate this, plugging $0$ into the cosine gives $$\cos(0) = 1 = \cos(0 + 2\pi) = \cos(0 + 4\pi) = \dots.$$