Positive function for which the sequence $\int_{\mathbb{R}}f(t)\frac{\sin^2((2n+1)t)}{\sin^2(t)}dt$ is bounded (on $n\in\mathbb{N}$).

Question

Is there an example of a "big" function $f:\mathbb{R}\to\mathbb{R}_+$ (say, not exponentially decaying at infinity) for which the sequence $$\int_{\mathbb{R}}f(t)\frac{\sin^2((2n+1)t)}{\sin^2(t)}dt$$ is bounded (on $n\in\mathbb{N}$)?