I need to sum up several Dirichlet kernels. To do this, I would like to have a compact formula for $$ \sum_{n=-N}^N \sin(2nx+\xi) $$ where $x,\xi \in \mathbb R$. The final result should look like something similar to a product of two Dirichlet Kernels.
To be more precise: I want to sum up $$ \sum_{n=-N}^N \sin( (M+n)2\phi ) $$ for $M \geq N$ a natural number and $\phi \in \mathbb R$.
Note that
$$\begin{align}\sum_{n=-N}^N \sin(2(n+M)\phi)&=\sin(2M\phi)\left(1+2\sum_{n=1}^N \cos(2n\phi)\right)\\\\ &=\sin(2M\phi)\left(1+2\text{Re}\left(\sum_{n=1}^N \left(e^{i2\phi}\right)^n\right)\right)\\\\ &=\sin(2M\phi)\left(1+2\text{Re}\left(\frac{e^{i2\phi}-e^{i2(N+1)\phi}}{1-e^{i2\phi}}\right)\right)\\\\ &=\sin(2M\phi)\left(1+2\frac{\cos((N+1)\phi)\sin(N\phi)}{\sin(\phi)}\right)\\\\ &=\frac{\sin(2M\phi)\sin((2N+1)\phi)}{\sin(\phi)} \end{align}$$
