Question

Помогите решить ................

Answer 1

1)

$sin(\alpha+\beta)=sin\alpha cos\beta+cos\alpha sin \beta$

$sin(\alpha-\beta)=sin\alpha cos\beta-cos\alpha sin \beta$

+___________

$sin(\alpha+\beta)+sin(\alpha-\beta)=2sin\alpha cos\beta$

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$cos\frac{2\pi}{7}+cos\frac{4\pi}{7}+cos\frac{6\pi}{7}=$

$\frac{2sin\frac{\pi}{7} \left(cos\frac{2\pi}{7}+cos\frac{4\pi}{7}+cos\frac{6\pi}{7} \right) }{2sin\frac{\pi}{7}}=$

$\frac{2sin\frac{\pi}{7} cos\frac{2\pi}{7}+2sin\frac{\pi}{7}cos\frac{4\pi}{7}+2sin\frac{\pi}{7}cos\frac{6\pi}{7}}{2sin\frac{\pi}{7}}=$

$\frac{sin \left(\frac{\pi}{7} +\frac{2\pi}{7}\right)+sin\left(\frac{\pi}{7} -\frac{2\pi}{7}\right) +sin \left(\frac{\pi}{7}+\frac{4\pi}{7} \right)+sin \left(\frac{\pi}{7}-\frac{4\pi}{7} \right)+sin \left(\frac{\pi}{7}+\frac{6\pi}{7} \right)+sin \left(\frac{\pi}{7}-\frac{6\pi}{7} \right) }{2sin\frac{\pi}{7}}=$

$\frac{sin\frac{3\pi}{7}+sin\left(-\frac{\pi}{7}\right) +sin \frac{5\pi}{7} +sin \left(-\frac{3\pi}{7} \right)+sin \pi+sin \left(-\frac{5\pi}{7} \right) }{2sin\frac{\pi}{7}}=$

$\frac{sin\frac{3\pi}{7}-sin\frac{\pi}{7}+sin\frac{5\pi}{7} -sin \frac{3\pi}{7} +sin \pi-sin\frac{5\pi}{7} }{2sin\frac{\pi}{7}}=$

$\frac{-sin\frac{\pi}{7} +sin \pi }{2sin\frac{\pi}{7}}=\frac{-sin\frac{\pi}{7} +0}{2sin\frac{\pi}{7}}=\frac{-sin\frac{\pi}{7} }{2sin\frac{\pi}{7}}=-\frac{1}{2}$

2)

$cos(\alpha+\beta)=cos\alpha cos\beta+sin\alpha sin \beta$

$cosn(\alpha-\beta)=cos\alpha cos\beta-sin\alpha sin \beta$

+___________

$cos(\alpha+\beta)+cos(\alpha-\beta)=2cos\alpha cos\beta$

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$cos\frac{2\pi}{7}cos\frac{4\pi}{7}+cos\frac{2\pi}{7}cos\frac{6\pi}{7}+cos\frac{4\pi}{7}cos\frac{6\pi}{7}=$

$\frac{2cos\frac{2\pi}{7}cos\frac{4\pi}{7}}{2}+\frac{2cos\frac{2\pi}{7}cos\frac{6\pi}{7}}{2}+\frac{2cos\frac{4\pi}{7}cos\frac{6\pi}{7}}{2}=$

$\frac{cos \left(\frac{2\pi}{7} +\frac{4\pi}{7} \right) +cos\left(\frac{2\pi}{7}-\frac{4\pi}{7} \right)}{2}+\frac{cos \left(\frac{2\pi}{7} +\frac{6\pi}{7} \right) +cos\left(\frac{2\pi}{7}-\frac{6\pi}{7} \right)}{2}+\frac{cos \left(\frac{4\pi}{7} +\frac{6\pi}{7} \right) +cos\left(\frac{4\pi}{7}-\frac{6\pi}{7} \right)}{2}=$

$\frac{cos \frac{6\pi}{7} +cos\left(-\frac{2\pi}{7} \right)}{2}+\frac{cos \frac{8\pi}{7} +cos\left(-\frac{4\pi}{7} \right)}{2}+\frac{cos \frac{10\pi}{7} +cos\left(-\frac{2\pi}{7} \right)}{2}=$

$\frac{cos \frac{6\pi}{7} +cos\frac{2\pi}{7}}{2}+\frac{cos \frac{8\pi}{7} +cos\frac{4\pi}{7} }{2}+\frac{cos \frac{10\pi}{7} +cos\frac{2\pi}{7}}{2}=$

$\frac{cos\frac{2\pi}{7}+cos\frac{4\pi}{7}+cos \frac{6\pi}{7} +cos\frac{2\pi}{7}+cos \frac{8\pi}{7} +cos \frac{10\pi}{7} }{2}=$

$\frac{cos\frac{2\pi}{7}+cos\frac{4\pi}{7}+cos \frac{6\pi}{7} +cos\frac{2\pi}{7}+cos \left(\pi+\frac{\pi}{7} \right) +cos \left( \pi+\frac{3\pi}{7}\right) }{2}=$

$\frac{cos\frac{2\pi}{7}+cos\frac{4\pi}{7}+cos \frac{6\pi}{7} +cos\frac{2\pi}{7}-cos \frac{\pi}{7} -cos \frac{3\pi}{7} }{2}=$

$\frac{-\frac{1}{2} +cos\frac{2\pi}{7}-cos \frac{\pi}{7} -cos \frac{3\pi}{7} }{2}=(*)$

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$sin(\alpha+\beta)=sin\alpha cos\beta+cos\alpha sin \beta$

$sin(\alpha-\beta)=sin\alpha cos\beta-cos\alpha sin \beta$

+___________

$sin(\alpha+\beta)+sin(\alpha-\beta)=2sin\alpha cos\beta$

$cos\frac{2\pi}{7}-cos \frac{\pi}{7} -cos \frac{3\pi}{7}=$

$\frac{2sin\frac{\pi}{7} \left( cos\frac{2\pi}{7}-cos \frac{\pi}{7} -cos \frac{3\pi}{7} \right) }{2sin\frac{\pi}{7}}=\frac{2sin\frac{\pi}{7} cos\frac{2\pi}{7}-2sin\frac{\pi}{7}cos \frac{\pi}{7} -2sin\frac{\pi}{7}cos \frac{3\pi}{7} }{2sin\frac{\pi}{7}}=$

$\frac{sin \left(\frac{\pi}{7} +\frac{2\pi}{7} \right)+sin\left(\frac{\pi}{7} -\frac{2\pi}{7} \right) -sin \left(\frac{\pi}{7}+\frac{\pi}{7} \right)-sin\left(\frac{\pi}{7}-\frac{\pi}{7} \right) -sin \left(\frac{\pi}{7}+ \frac{3\pi}{7} \right) -sin\left(\frac{\pi}{7}- \frac{3\pi}{7} \right) }{2sin\frac{\pi}{7}}=$

$\frac{sin\frac{3\pi}{7} +sin\left(-\frac{\pi}{7} \right) -sin \frac{2\pi}{7} -sin 0 -sin \frac{4\pi}{7} -sin\left(- \frac{2\pi}{7} \right) }{2sin\frac{\pi}{7}}=$

$\frac{sin\frac{3\pi}{7} -sin\frac{\pi}{7} -sin \frac{2\pi}{7} -0 -sin \frac{4\pi}{7} +sin \frac{2\pi}{7} }{2sin\frac{\pi}{7}}=$

$\frac{sin\frac{3\pi}{7} -sin \frac{\pi}{7} -sin \frac{4\pi}{7} }{2sin\frac{\pi}{7}}=\frac{sin\frac{3\pi}{7} -sin \frac{4\pi}{7} -sin \frac{\pi}{7} }{2sin\frac{\pi}{7}}=$

$\frac{2cos\frac{\frac{3\pi}{7}+\frac{4\pi}{7} }{2}sin\frac{\frac{3\pi}{7}-\frac{4\pi}{7} }{2} -sin \frac{\pi}{7} }{2sin\frac{\pi}{7}}=$

$\frac{2cos\frac{\pi}{2}sin \left(\frac{\pi}{14}\right) -sin \frac{\pi}{7} }{2sin\frac{\pi}{7}}=$

$\frac{0 -sin \frac{\pi}{7} }{2sin\frac{\pi}{7}}=\frac{ -sin \frac{\pi}{7} }{2sin\frac{\pi}{7}}=-\frac{1}{2}$

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$(*)=\frac{-\frac{1}{2} -\frac{1}{2} }{2}=-\frac{1}{2}$