To give some background, I am trying to solve this Determine if this is a subring. problem. I have determined that, if I can show that the statement in yellow is false, i.e. contradictory, my proof should be complete.
$\frac12 sin(2t) \equiv b_1cos(t)+...+b_kcos(kt)+c_1sin(t)+...c_lsin(lt)$
where $b_1,...,b_k,c_1,...,c_l\in \mathbb Z$
My thought is that a contradiction might be obtainaned by either taking integrals or derivatives of both sides, but so far I have not been very successful. Please help me!
Please keep the following in mind when answering:
I do not know Fourier Analysis
I do not know Complex Analysis
I am still a beginner at Ring Theory
I am looking for a solution that is easy (for me) to understand