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Show that for every \(n \in \mathbb{N}\), $$ \mathrm{Vect}\big( x \mapsto \cos(kx) \big)_{0 \le k \le n} = \mathrm{Vect}\big( x \mapsto \cos^k x \big)_{0 \le k \le n}. $$ Hint. Use the trigonometric recurrence $$ \cos((k+1) x) = 2 \cos x \cos(k x) - \cos((k - 1) x). $$
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