Summation!

Say $S\,=\,\large \sum_{m=1}^{6}\csc\left(\omega+\frac{(m-1)\pi}{4})\right)\csc\left(\omega+\frac{m\pi}{4}\right)$ . Plugging in the values of $m$ and then writing all the six terms :- $T_1\,=\,\frac{\sqrt2}{\sin(\omega)[\sin(\omega)+\cos(\omega)]}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,T_2\,=\,\frac{\sqrt2}{\cos(\omega)[\sin(\omega)+\cos(\omega)]} \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, T_3\,=\,\frac{\sqrt2}{\cos(\omega)[\cos(\omega)-\sin(\omega)]}$
$T_4\,=\,\frac{\sqrt2}{\sin(\omega)[\sin(\omega)-\cos(\omega)]} \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, T_5\,=\,\frac{\sqrt2}{\sin(\omega)[\sin(\omega)+\cos(\omega)]}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,T_6\,=\,\frac{\sqrt2}{\cos(\omega)[\sin(\omega)+\cos(\omega)]}$ Now, $S\,=\, \sum_{i=1}^{6}T_i \, = (T_1\,+\,T_4) \,+\, (T_2\,+\,T_4) \,+\, (T_5\,+\,T_6) \,= \, \left ( \left ( -\frac{2\sqrt2}{\cos(2\omega)} \right ) \,+\, \left ( \frac{2\sqrt2}{\cos(2\omega)} \right ) \,+\, \left ( \frac{2\sqrt2}{\sin(2\omega)} \right ) \right ) \, \Rightarrow\ S\,=\,\frac{2\sqrt2}{\sin(2\omega)} \Rightarrow\ 4\sqrt2 \,=\frac{2\sqrt2}{\sin(2\omega)} \Rightarrow\sin(2\omega)=\frac{1}{2} \Rightarrow\omega\,=\,\frac{\pi}{12}$

Now, $\sum_{r=1}^{7}\tan^{2}\left (\frac{3r\omega}{4}\right )\,=\,\sum_{r=1}^{7}\tan^{2}\left (r \frac{\pi}{16} \right )\,=\, \sum_{r=1}^{7}\frac{1-\cos(\frac{r\pi}{8})}{1+\cos(\frac{r\pi}{8})}$

$=\left ( \underbrace{\left ( \frac{1-\cos(\frac{\pi}{8})}{1+\cos(\frac{\pi}{8})}\,+\, \frac{1-\cos(\frac{3\pi}{8})}{1+\cos(\frac{3\pi}{8})} \,+\, \frac{1-\cos(5\frac{\pi}{8})}{1+\cos(5\frac{\pi}{8})} \,+ \frac{1-\cos(\frac{7\pi}{8})}{1+\cos(\frac{7\pi}{8})} \right )}_\text{Adding these four terms and further simplifying gives 28} \right ) \,+\, \left ( \underbrace{\left ( \frac{1-\cos(\frac{2\pi}{8})}{1+\cos(\frac{2\pi}{8})}\,+\, \frac{1-\cos(\frac{4\pi}{8})}{1+\cos(\frac{4\pi}{8})} \,+\, \frac{1-\cos(\frac{6\pi}{8})}{1+\cos(\frac{6\pi}{8})} \right )}_{7} \right )$

$28\,+\,7\,=\,35$