Suppose $$T_n=\{p(t)=\sum_{k=0}^{n}a_k\cos(kt)+\sum_{k=1}^{n}b_k\sin(kt)\mid a_k,b_k\in\mathbb{C}\}$$
$$ L_k(t)=\prod_{\substack{i=0\\i\ne k}}^{2n}\frac{\sin \frac{t-t_i}{2}}{\sin \frac{t_k-t_i}{2}} \hspace{6mm}, (k=0,1,\dots,2n) $$
I am trying to show that $L_k(t) \in T_n$
I tried to manipulate the series with $e^{it}=\cos t + i\sin t$ to somehow prove it but could not proceed.