Prove that {$\sin(x) , \sin(2x) , \sin(3x) ,...,\sin(nx)$} is independent in $\mathbb{R}$
my trial :
we know that the Wronsekian shouldn't be $0$ to get the trivial solution and thus they are independent. its not trivial to show that $ W \not = 0$
W =
$\begin{vmatrix} (1)\sin(x) & (1)\sin(2x) & (1)\sin(3x) & ... & (1)\sin(nx) \\ (1)\cos(x) & (2)\cos(2x) & (3)\cos(3x) & ... & (n)\cos(nx) \\ -(1)^2\sin(x) & -(2)^2\sin(2x) & -(3)^2\sin(3x) & ... & -(n)^2\sin(nx) \\ -(1)^3\cos(x) & -(2)^3\cos(2x) & -(3)^3\cos(3x) & ... & -(n)^3\cos(nx) \\ \end{vmatrix}$
and so on. it looks like Vandermonde matrix but i cant prove that and so we conclude that its $W\not =0$