linear algebra - Is the subset ${f_n: ngeq 0}$ of $mathbb{R}^mathbb{R}$ linearly independent.

For each non-negative integer $n$, let $f_n\in \mathbb{R}^\mathbb{R}$ be the function defined by $f_n$: $x\to \sin^n(x)$. Is the subset $\{f_n: n\geq 0\}$ of $\mathbb{R}^\mathbb{R}$ linearly independent.

So I did an induction proof for this. However, I feel that this is the incorrect method for a problem like this because we need to show that any subset of the following form is linearly independent. How can I show that this set is linearly independent? Any solutions or hints are greatly appreciated.

Base case: When $n=0$ we have $\sin^0(x)=1$ and a linear combination of a non-zero vector is always L.I.. So the base case is satisfied. Assume the result holds for some $k