1. ## Proof

Let t be a fixed number and let $c=cos(t)$ and $s=sin(t)$ and $\mathbf{x}=(c, cs, cs^2,..., cs^{n-1}, s^n)^T$.

Show that $\mathbf{x}$ is a unit vector.

Hint: $1+s^2+s^4+...+s^{2n-2}=\frac{1-s^{2n}}{1-s^2}$

Not sure how that even helps unless I want to the hint via induction.

2. Originally Posted by dwsmith
Let t be a fixed number and let $c=cos(t)$ and $s=sin(t)$ and $\mathbf{x}=(c, cs, cs^2,..., cs^{n-1}, s^n)^T$.

Show that $\mathbf{x}$ is a unit vector.

Hint: $1+s^2+s^4+...+s^{2n-2}=\frac{1-s^{2n}}{1-s^2}$

Not sure how that even helps unless I want to the hint via induction.
Yes, sure it helps, for the square of the length of that vector is
$c^2+c^2s^2+\cdots+c^2s^{2n-2}+s^{2n}=c^2\frac{1-s^{2n}}{1-s^2}+s^{2n}=1-s^{2n}+s^{2n}=1$
since $1-s^2=c^2$.

3. Originally Posted by Failure
Yes, sure it helps, for the square of the length of that vector is
$c^2+c^2s^2+\cdots+c^2s^{2n-2}+s^{2n}=c^2\frac{1-s^{2n}}{1-s^2}+s^{2n}=1-s^{2n}+s^{2n}=1$
since $1-s^2=c^2$.
How did you get $c^2$?

4. Originally Posted by dwsmith
How did you get $c^2$?
Sorry, but I don't quite understand your question.
I just wrote down what, imho, the square of the length of that vector $(c,cs,cs^2,\ldots,cs^{n-1},s^n)$ happens do be: namely the sum of the squares of its coordinates. Hence I get
$c^2+(cs)^2+\cdots+(cs^{n-1})^2+(s^n)^2=c^2+c^2s^2+\cdots+c^2s^{2n-2}+s^{2n}$
Can you be a bit more specific as to where, exactly the occurrence of $c^2$ in what I wrote surprises you?

5. If you are referring to his " $1- s^2= c^2$, remember that you said "let c= cos(t) and s= sin(t)". Failure (he really should change that user name!) is just using the fact that $sin^2(t)+ cos^2(t)= 1$.