Spectrum of the integral operator $A(f)(x)=\int_{[0,2\pi]}\frac{\sin(n\frac{x-y}{2})}{\sin(\frac{x-y}{2})}f(y)dy$ where $A:L^2(0,2\pi)\to L^2(0,2\pi)$

Question

I want to understand the spectrum of the integral operator $A$ from $L^2(0,2\pi)$ to itself, given by $$A(f)(x):=\int_{[0,2\pi]}\frac{\sin(n\frac{x-y}{2})}{\sin(\frac{x-y}{2})}f(y)dy$$ where $n$ is a positive integer. I want to show that the spectrum is contained in $[0,1]$. (Eventually I want to show that spectrum lies in the same interval even if we define the operator by taking the integral on any set $B\subseteq [0,2\pi]$, but I guess it's better to start with this case). But I couldn't figure out how to handle this. I'm not good at Fourier analysis. I've tried using the formula $\sum_{k=0}^{n-1}e^{ikx} =e^{i(n-1)x/2}\frac{\sin(n\frac{x}{2})}{\sin(\frac{x}{2})}$ but couldn't conclude anything. I'd be glad for any hint, any idea about how to approach/understand it.