I would like to determine under what conditions on $k$ the set $ \begin{align} A = &\{1,\cos(t),\sin(t), \\ &\quad \cos(t(1+k)),\sin(t(1+k)),\cos(t(1−k)),\sin(t(1−k)), \\ &\quad \cos(t(1+2k)),\sin(t(1+2k)),\cos(t(1−2k)),\sin(t(1−2k))\}, \end{align}$ is linearly independent, where $k$ is some arbitrary real number.
As motivation, I know that the set defined by
$ \{1, \cos wt, \sin wt\}, \quad w = 1, \dots, n $
is linearly independent on $\mathbb{R}$, which one generally proves by computing the Wronskian. I thought that I could extend this result to the set in question, but I haven't found a proper way to do so. My intuition tells me that $A$ will be linearly dependent when the arguments of the trig functions coincide, which will depend on the value of $k$.
Though, I'm at a loss for proving this is true. Computing the Wronskian for this set required an inordinate amount of time-- I stopped running the calculation after a day. Is there perhaps a way to reduce the set in question so that the Wronskian becomes manageable?
I'm interested in any suggestions/alternative methods for proving linear independence that could help my situation. Note that I'd like to have a result that holds for any $m = 0, \dots, n,$ where $n \in \mathbb{Z}$ if possible.
Thanks for your time.
EDIT: The set originally defined in the first instance of this post was incorrectly cited. My sincere apologies.