$$ \begin{aligned} & \text { If } \mathrm{a}=\sin \frac{\pi}{7}+\sin \frac{2 \pi}{7}+\sin \frac{3 \pi}{7}, \mathrm{~b}=\cos \frac{\pi}{7}+\cos \frac{2 \pi}{7}+\cos \frac{3 \pi}{7}, \mathrm{c}=\tan \frac{\pi}{7}+\tan \frac{2 \pi}{7}+\tan \frac{3 \pi}{7} \\ & \text { then }\left|\begin{array}{ccc} 6 a^2+10 b^2 & 8 a b+25 b c & 10 a c+30 a b \\ 9 a b-8 b c & 12 b^2-20 c^2 & 15 b c-24 a c \\ 12 a c-10 a b & 16 b c-25 a c & 20 c^2-30 a^2 \end{array}\right| \text { equals } \\ & \end{aligned} $$
So far I have been to find values of a, b & c like this $$ \begin{aligned} &\begin{aligned} a&=\sin \frac{\pi}{7}+\sin \frac{2 \pi}{7}+\sin \frac{3 \pi}{7} \\ & =\frac{\sin \left(\frac{3 \pi}{14}\right)}{\sin \left(\frac{\pi}{14}\right)} \sin \left(\frac{\pi}{7}+\frac{\pi}{7}\right) \\ & =\frac{\sin \left(\frac{3 \pi}{14}\right) \sin \left(\frac{2 \pi}{7}\right)}{\sin \frac{\pi}{14}}=\frac{\sin \frac{3 \pi}{14} \cdot \cos \left(\frac{3 \pi}{14}\right)}{\sin \left(\frac{\pi}{14}\right)} \\ & =\frac{\sin \left(\frac{6 \pi}{14}\right)}{2 \sin \frac{\pi}{14}}=\frac{1}{2} \cot \frac{\pi}{7} \\ b&=\cos \frac{\pi}{7}+\cos \frac{2 \pi}{7}+\cos \frac{3 \pi}{7}=\frac{\cos \left(\frac{3 \pi}{1 m}\right) \cos \left(\frac{2 \pi}{7}\right)}{\cos \frac{\pi}{7}}=\frac{1}{2} \frac{\sin \frac{6 \pi}{7}}{\cos \frac{\pi}{7}}=\frac{1}{2} \tan \frac{\pi}{7} \\ & \end{aligned}\\ &\begin{aligned} c&=\tan \frac{\pi}{7}+\tan \frac{2 \pi}{7}+\tan \frac{3 \pi}{7}=\sin \frac{\pi}{7} \cos \frac{2 \pi}{7} \cos \frac{3 \pi}{7}+\cos \frac{\pi}{7} \cdot \sin \frac{2 \pi}{7} \cdot \cos \frac{3 \pi}{7} \\ & +\cos \frac{\pi}{7} \cdot \cos \frac{2 \pi}{7} \frac{\sin 3 \pi}{7} \\ & \cos \frac{\pi}{7} \cdot \cos \frac{2 \pi}{7} \cdot \cos \frac{3 \pi}{7} \\ & =\frac{\sin \left(\frac{\pi}{7}+\frac{2 \pi}{7}+\frac{3 \pi}{7}\right)+\sin \frac{\pi}{7} \cdot \sin \frac{2 \pi}{7} \cdot \sin \frac{3 \pi}{7}}{\cos \frac{\pi}{7} \cdot \cos \frac{2 \pi}{7} \cos \frac{3 \pi}{7}} \\ & =\frac{\sin \frac{\pi}{7}+\sin \frac{\pi}{7} \cdot \sin \frac{2 \pi}{7} \cdot \sin \frac{3 \pi}{7}}{\cos \frac{\pi}{7} \cdot \cos \frac{2 \pi}{7} \cdot \cos \frac{3 \pi}{7}} \\ & \end{aligned} \end{aligned} $$