\[\begin{array}{l} {\text{Докажем}}{\text{, что числитель равен знаменателю}}{\text{.}} \hfill \\ \operatorname{tg} \frac{{3\pi }}{7} - 4\sin \frac{\pi }{7} = \operatorname{tg} \frac{{6\pi }}{7} + 4\sin \frac{{2\pi }}{7} \Leftrightarrow \operatorname{tg} \frac{{3\pi }}{7} - \operatorname{tg} \frac{{6\pi }}{7} = 4\left( {\sin \frac{\pi }{7} + \sin \frac{{2\pi }}{7}} \right) \Leftrightarrow \hfill \\ \operatorname{tg} \frac{{3\pi }}{7} + \operatorname{tg} \frac{\pi }{7} = 4\left( {\sin \frac{\pi }{7} + \sin \frac{{2\pi }}{7}} \right) \hfill \\ \operatorname{tg} x + \operatorname{tg} y = \frac{{\sin \left( {x + y} \right)}}{{\cos x\cos y}} \hfill \\ \sin x + \sin y = 2\sin \frac{{x + y}}{2}\cos \frac{{x - y}}{2} \hfill \\ \frac{{\sin \frac{{4\pi }}{7}}}{{\cos \frac{{3\pi }}{7}\cos \frac{\pi }{7}}} = 8 \cdot \sin \frac{{3\pi }}{{14}}\cos \frac{\pi }{{14}} \Leftrightarrow 2\sin \frac{{2\pi }}{7}\cos \frac{{2\pi }}{7} = 8 \cdot \sin \frac{{3\pi }}{{14}}\cos \frac{\pi }{{14}}\cos \frac{{6\pi }}{{14}}\cos \frac{{2\pi }}{{14}} \Leftrightarrow \hfill \\ \left[ {\sin \frac{{3\pi }}{{14}} = \cos \frac{{2\pi }}{7}} \right] \Leftrightarrow \sin \frac{{2\pi }}{7}\cos \frac{{2\pi }}{7} = 4 \cdot \cos \frac{{2\pi }}{7}\cos \frac{\pi }{{14}}\cos \frac{{6\pi }}{{14}}\cos \frac{{2\pi }}{{14}} \Leftrightarrow \hfill \\ \sin \frac{{2\pi }}{7} = 4 \cdot \cos \frac{\pi }{{14}}\cos \frac{{6\pi }}{{14}}\cos \frac{{2\pi }}{{14}} \Leftrightarrow \sin \frac{{2\pi }}{7}\sin \frac{\pi }{{14}} = 4 \cdot \sin \frac{\pi }{{14}}\cos \frac{\pi }{{14}}\cos \frac{{6\pi }}{{14}}\cos \frac{{2\pi }}{{14}} \Leftrightarrow \hfill \\ \sin \frac{{2\pi }}{7}\sin \frac{\pi }{{14}} = \left( {2 \cdot \sin \frac{{2\pi }}{{14}}\cos \frac{{2\pi }}{{14}}} \right)\cos \frac{{3\pi }}{7} \Leftrightarrow \sin \frac{{2\pi }}{7}\sin \frac{\pi }{{14}} = \sin \frac{{2\pi }}{7}\cos \frac{{3\pi }}{7} \Leftrightarrow \hfill \\ \sin \frac{\pi }{{14}} = \cos \frac{{3\pi }}{7} \Leftrightarrow \sin \frac{\pi }{{14}} = \sin \frac{\pi }{{14}}. \hfill \\ \end{array}\]

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