Question

Помогите решить ................

Answer 1

1)

$sin(alpha+beta)=sinalpha cosbeta+cosalpha sin beta$

$sin(alpha-beta)=sinalpha cosbeta-cosalpha sin beta$

+___________

$sin(alpha+beta)+sin(alpha-beta)=2sinalpha cosbeta$

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$cosfrac{2pi}{7}+cosfrac{4pi}{7}+cosfrac{6pi}{7}=$

$frac{2sinfrac{pi}{7} left(cosfrac{2pi}{7}+cosfrac{4pi}{7}+cosfrac{6pi}{7} right) }{2sinfrac{pi}{7}}=$

$frac{2sinfrac{pi}{7} cosfrac{2pi}{7}+2sinfrac{pi}{7}cosfrac{4pi}{7}+2sinfrac{pi}{7}cosfrac{6pi}{7}}{2sinfrac{pi}{7}}=$

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

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

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

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

2)

$cos(alpha+beta)=cosalpha cosbeta+sinalpha sin beta$

$cosn(alpha-beta)=cosalpha cosbeta-sinalpha sin beta$

+___________

$cos(alpha+beta)+cos(alpha-beta)=2cosalpha cosbeta$

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$cosfrac{2pi}{7}cosfrac{4pi}{7}+cosfrac{2pi}{7}cosfrac{6pi}{7}+cosfrac{4pi}{7}cosfrac{6pi}{7}=$

$frac{2cosfrac{2pi}{7}cosfrac{4pi}{7}}{2}+frac{2cosfrac{2pi}{7}cosfrac{6pi}{7}}{2}+frac{2cosfrac{4pi}{7}cosfrac{6pi}{7}}{2}=$

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

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

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

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

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

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

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

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$sin(alpha+beta)=sinalpha cosbeta+cosalpha sin beta$

$sin(alpha-beta)=sinalpha cosbeta-cosalpha sin beta$

+___________

$sin(alpha+beta)+sin(alpha-beta)=2sinalpha cosbeta$

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

$frac{2sinfrac{pi}{7} left( cosfrac{2pi}{7}-cos frac{pi}{7} -cos frac{3pi}{7} right) }{2sinfrac{pi}{7}}=frac{2sinfrac{pi}{7} cosfrac{2pi}{7}-2sinfrac{pi}{7}cos frac{pi}{7} -2sinfrac{pi}{7}cos frac{3pi}{7} }{2sinfrac{pi}{7}}=$

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

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

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

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

$frac{2cosfrac{frac{3pi}{7}+frac{4pi}{7} }{2}sinfrac{frac{3pi}{7}-frac{4pi}{7} }{2} -sin frac{pi}{7} }{2sinfrac{pi}{7}}=$

$frac{2cosfrac{pi}{2}sin left(frac{pi}{14}right) -sin frac{pi}{7} }{2sinfrac{pi}{7}}=$

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

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