linear algebra - Are the functions $sin^n(x)$ linearly independent?

The following problem is from Golan's linear algebra book. I have posted a proposed solution in the answers.

Problem: For $n\in \mathbb{N}$, consider the function $f_n(x)=\sin^n(x)$ as an element of the vector space $\mathbb{R}^\mathbb{R}$ over $\mathbb{R}$. Is the subset $\{f_n:\ n\in\mathbb{N}\}$ linearly independent?

Answer

Suppose we have real numbers $a_j$ such that $\sum_1^k a_j \sin^j(x)=0$ for every real $x$. Consider the polynomial $f(y)=\sum_1^k a_j y^j$. By assumption, we know that $f(\sin(x))=0$ for every $x$. Since $\sin(x)$ can take any value between $-1$ and $1$, we have that $f(y)=0$ for any $y$ between $-1$ and $1$. But then $f(y)=0$ for infinitely many values of $y$, and so $f$ is the zero polynomial, i.e. $a_j=0$ for all $j$. Thus the only linear dependence is the trivial one, and so our set is linearly independent.