Do trigonometric polynomials form an algebra

Question

Let a trigonometric polynomial be a function defined as $$a_0 + \sum_{k=0}^n \left[a_k \cos(kx) + b_k \sin(kx)\right].$$

Clearly the set of all such functions forms a vector space. Is it true that they form also an algebra?

Answer 1

By linearity, it suffices to check that the products $$\cos(kx) \cos(lx), \cos(kx) \sin(lx), \sin(kx) \sin(lx)$$ can all be written in this form, but this follows immediately from the usual product-to-sum identities and the symmetry properties of $\sin$ and $\cos$. For example, $$\cos(kx) \cos(lx) = \tfrac{1}{2}\left(\cos[(k + l) x] + \cos [(k - l) x]\right) .$$

Answer 2

Yes. Use

$\sin (x+y)=\sin x\cos y+\cos x\sin y$,

$\cos (x+y)=\cos x\cos y-\sin x\sin y$

for a proof.