Sum of Dirichlet kernel for angle differences over $n$ angles on unit circle

Let $$D(\theta_i-\theta_j):= \frac{\sin((n+1/2)(\theta_i-\theta_j))}{2\sin\frac{\theta_i-\theta_j}{2}}$$ being the Dirichlet type of kernel of angle difference between $\theta_i$ and $\theta_j$ where $\theta_i\in[0,2\pi]$ for all $i\in[n]$.

I am curious about how to estimate the lower bound of the sum of Dirichlet kernels over all pair of angles

$$\sum_{i=1}^n\sum_{j=1}^n \frac{\sin((n+1/2)(\theta_i-\theta_j))}{2\sin\frac{\theta_i-\theta_j}{2}}$$

Would it be possible to obtain its lower bound explicitly, if not, could we have its estimation that is larger than $n^2/2$ at least?