In Dummit and Foote , in section 1.2 Dihedral groups these are left as exercise.
We leave the details of the following calculations as an exercise
$(i)$ $1, r, r^2,...,r^{n-1}$ are all distinct and $r^{n} = 1$, so $|r| = n$.
$(ii)$ $|s|= 2$.
$(iii)$ $s \neq r^i$ for any $i$.
$(iv)$ $sr^i \neq sr^j$, for all $0 \leq i, j \leq n- 1$ with $i\neq j$, so i.e., each element can be written uniquely in the form $s^kr^i$ for some $k = 0$ or $1$ and $0 \leq i \leq n - 1$.
$(i)$ At first I want to show $r^n=1$ and $s^2=1$. But I couldn't. Because I know $r$ is the rotation about the origin $\frac{2\pi}{n}$. Now after rotatating n times we rotate $n×\frac{2\pi}{n}=2\pi$ times. And we got $1$. So $r^n=1$ and similarly for refelction after rotatating 2 times we get the 1 so $s^2=1$.but this is intuitive. I can't find a way to show this mathematically, I think for that I need mathematical representation of $r$.
Once we can show $r^n=s^2=1$ then rest are very easy.for $(i)$ if $r^i=r^j$ then $r^{i-j}=1$ so $i-j \geq n$ but as $0 \leq i, j \leq n- 1$ so $i-j \leq n-1 \lt n$ so contradiction. And for $(iii)$ and for first part of $(iv)$ proceeding same as $(i)$ we can get contradiction.
But how to show the uniqueness part? What is the meaning of each element? Do we have to show all the powers $r,s$ that is $s^pr^q$, $p,q \in \mathbb{Z}$ can be written in the form $s^kr^i$, where $k,i$ is as defined in the question $(iv)$, but thats follows from $r^n=s^2=1$ so , its reduced to $s^{p \pmod{2}} r^{q \pmod{n}}$ , so if let $k=p \pmod{2}$ and $i=q \pmod{n}$ then we have desired result. But how to show its unique?