\( \definecolor{colordef}{RGB}{249,49,84} \definecolor{colorprop}{RGB}{18,102,241} \)
In the vector space \(C(\mathbb{R}, \mathbb{R})\), set \(F = \mathrm{Vect}(\sin)\) and \(G = \mathrm{Vect}(\cos)\). Show that the sum \(F + G\) is exactly the set of functions of the form \(x \mapsto a \cos x + b \sin x\) with \(a, b \in \mathbb{R}\).
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