TITLE: Are the functions $\sin^n(x)$ linearly independent?
QUESTION [13 upvotes]: 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?

REPLY [4 votes]: The Wronskian of $\sin(t),...,\sin^n(t)$ is
$$
1! 2! 3! \dots (n-1)! \sin^n(t) \cos^{n(n-1)/2}(t)
$$
not identically zero.