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I have been asked to prove that $\{\tan(ax)|a\in \mathbb{R}^+\}$ is linearly independent. I was wondering if there is a generic method/idea for proving linear independence of functions over $\mathbb{R}$.

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Hint: Since $\tan \in C^{\infty}(\mathbf{R})$. Pick any finite number of $a_i$'s from $\mathbf{R}$. Show the Wronskian of $\tan{a_1x},\dots,\tan{a_nx}$ is nonzero.

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