Show this is contradictory: $\frac12 sin(2t) \equiv b_1cos(t)+...+b_kcos(kt)+c_1sin(t)+...c_lsin(lt)$ where $b_1,...,b_k,c_1,...,c_l\in \mathbb Z$

Question

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:

1. I do not know Fourier Analysis

2. I do not know Complex Analysis

3. I am still a beginner at Ring Theory

4. I am looking for a solution that is easy (for me) to understand

Answer 1

Hint: Suppose this is possible. Set $t=\frac\pi4$ and show this implies$\frac12$ is an integer combination of $1, \frac1{\sqrt2}$, which leads to a contradiction.