Recall that real trigonometric polynomials are functions of the form $$
f(\theta) = \frac{a_0}{2} +
\sum_{k=1}^n (a_k \cos(k\theta) + b_k \sin(k\theta)).
$$
I want to prove that real trigonometric polynomials are closed under multiplication.
In other words, given two trigonometric polynomials $f$ and $g$, I want to show that the function $h(\theta) = f(\theta)\cdot g(\theta)$
is also a trigonometric polynomial.
Toward this end I've noticed the double-angle formulas $$
\sin(2\theta) = 2\sin(\theta)\,\cos(\theta) \\
\cos(2\theta) = \cos^2(\theta) – \sin^2(\theta)
$$
might be helpful for the $k=2$ case. But I'm not sure what to do for terms with $k>2$ and I'm not sure how to handle the coefficients $a_k$ and $b_k$.
Any pointers are greatly appreciated.
Best Answer
$$2\sin \,A \sin \, B =\cos (A-B) -\cos (A+B),\\ 2cos \,A \cos \, B =\cos (A-B) +\cos (A+B), \\ 2\sin \,A \cos \, B =\sin (A+B)+\sin (A-B),\\ 2\cos \,A \sin \, B =\sin (B+A) -\sin (B-A).$$