Упростите выражения:а)(sin36°+sin40°+sin44°+sin48°)/(2sin88°cos4°sin42°)б)(cos6°+cos12°+cos36°+cos42°)/(sin87°cos15°cos24°)в)(cos16°-cos24°-cos32°+cos40°)/(cos86°sin8°cos28°)г)(sin48°-sin60°-sin72°+sin84°)/(4cos84°sin12°sin66°)

Воспользуемся формулами преобразования суммы в произведение: [latex]sinx+siny=2sin \frac{x+y}{2}cos \frac{x-y}{2} [/latex] [latex]cosx-cosy=-2sin \frac{x+y}{2}sin \frac{x-y}{2} [/latex] [latex]cosx+cosy=2cos \frac{x+y}{2}cos \frac{x-y}{2} [/latex] a) [latex] \frac{sin36к+sin40к+sin44к+sin48к}{2sin88кcos4кsin42к} = \frac{(sin36к+sin48к)+(sin40к+sin44к)}{2sin88кcos4кsin42к} = [/latex][latex]=\frac{2sin42кcos6+2sin42кcos2к}{2sin88кcos4кsin42к}= \frac{2sin42к(cos6+cos2к)}{2sin88кcos4кsin42к}=\frac{cos6+cos2к}{sin88к*cos4к}=[/latex][latex]= \frac{2cos4к*cos2к}{2sin88к*cos4к}= \frac{cos(90к-88к)}{sin88к} = \frac{sin88к}{sin88к}=1 [/latex] [latex]sin36к+sin48к=2sin \frac{36к+48к}{2}cos \frac{36к-48к}{2} =2sin42кcos6к [/latex] [latex]sin40к+sin44к=2sin \frac{40к+44к}{2}cos \frac{40к-44к}{2} =2sin42кcos2к [/latex] [latex]cos6к+cos2к=2cos \frac{6к+2к}{2}cos \frac{6к-2к}{2} =2cos4кcos2к [/latex] б) [latex] \frac{cos6к+cos12к+cos36к+cos42к}{sin87кcos15кcos24к} = \frac{(cos6к+cos42к)+(cos12к+cos36к)}{sin87кcos15кcos24к} = [/latex][latex]=\frac{2cos24кcos18к+2cos24кcos12к}{sin87кcos15кcos24к}=\frac{2cos24к(cos18к+cos12к)}{sin87кcos15кcos24к}=[/latex][latex]=\frac{2(cos18к+cos12к)}{sin87кcos15к}= \frac{2*2cos15кcos3к}{sin87к*cos15к} = \frac{4cos3к}{cos(90к-3к)} = \frac{4cos3к}{cos3к}=4 [/latex] в) [latex] \frac{cos16к-cos24к-cos32к+cos40к}{cos86кsin8кcos28к} = \frac{(cos16к+cos40к)-(cos24к+cos32к)}{cos86кsin8кcos28к} = [/latex][latex]=\frac{2cos28кcos12к-2cos28кcos4к}{cos86кsin8кcos28к} = \frac{2cos28к(cos12к-cos4к)}{cos86кsin8кcos28к} =\frac{2(cos12к-cos4к)}{cos86кsin8к} =[/latex][latex]=\frac{2*(-2sin8кsin4к)}{cos86кsin8к} = \frac{-4sin4к}{cos(90к-4к)}= \frac{-4sin4к}{sin4к}=-4[/latex] г) [latex]\frac{sin48к-sin60к-sin72к+sin84к}{4cos84кsin12кsin66к} = \frac{(sin48к+sin84к)-(sin60к+sin72к)}{4cos84кsin12кsin66к}=[/latex][latex]=\frac{2sin66кcos18к-2sin66кcos6к}{4cos84кsin12кsin66к}=\frac{2sin66к(cos18к-cos6к)}{4cos84кsin12кsin66к}=[/latex][latex]=\frac{cos18к-cos6к}{2cos84кsin12к}=\frac{cos18к-cos6к}{2cos84кsin12к}=\frac{-2sin12кsin6к}{2cos84кsin12к}=\frac{-sin6к}{cos84к}=[/latex][latex]=\frac{-sin6к}{cos(90к-6к)}=\frac{-sin6к}{sin6к}=-1[/latex]