Basis of trigonometric functions
The problem is as follows:
Let $C[a,b]$ be the vector space (over $\mathbb{R}$) of continuous functions defined on $[a,b]$, $a < b$. For a fixed $n \in \mathbb{Z}^+$, consider the following set of functions: $$ X_1 = \{1,\cos(x),\sin(x),\cos(2x),\sin(2x),\dots,\cos(nx),\sin(nx)\} \\ X_2 = \{1,\cos(x),\sin(x),\cos^2(x),\sin^2(x),\dots,\cos^n(x),\sin^n(x)\} $$
- Show that both $X_1$ and $X_2$ are linearly independent.
- Show that $\mathrm{span}(X_1) = \mathrm{span}(X_2)$.
- Let $V := \mathrm{span}(X_1)$. For each $v \in V$, let $[v]_1$ and $[v_2]$ be the coordinate vectors of $v$ with respect to the bases $X_1$ and $X_2$ respectively. Do there exists $v \in V$ such that $[v]_1 = [v]_2$, and $[v]_1 \neq (a_0,a_1,a_2,0,\dots,0)$ where $a_0, a_1, a_2 \in \mathbb{R}$?
My attempt to this problem were mostly futile, with virtually no progress for Q1 whatsoever. Q2 seems to be the easiest problem among all. One can easily show that the statement holds for $n \leq 3$ using well-known trigonometric identities. I strongly believe that I need to argue inductively, but I'm not exactly sure how. Lastly, I have absolutely no idea how to start for Q3.
Any help would be appreciated.
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