12.4积化和差与和差化积公式课后篇巩固提升基础达标练1.已知cosα-cosβ=12,sinα-sinβ=-13,则tanα+β2=.解析因为cosα-cosβ=12,所以-2sinα+β2sinα-β2=12,因为sinα-sinβ=-13,所以2cosα+β2sinα-β2=-13,因为sinα-β2≠0,cosα+β2≠0,所以-tanα+β2=-32,即tanα+β2=32.答案322.把tanx-tany化为积的形式为()A.cos\(x-y\)cosxcosyB.sin\(x+y\)sinxcosyC.sin\(x-y\)cosxcosyD.cos\(x+y\)cosxsiny解析tanx-tany=sinxcosx−sinycosy=sinxcosy-sinycosxcosxcosy=12[sin\(x+y\)+sin\(x-y\)]-12[sin\(y+x\)+sin\(y-x\)]cosxcosy2=sin\(x-y\)cosxcosy.答案C3.cos20°-cos50°=()A.cos35°cos15°B.sin35°sin15°C.2sin15°sin35°D.2sin15°cos35°解析cos20°-cos50°=-2sin20°+50°2sin20°-50°2=-2sin35°sin(-15°)=2sin15°sin35°.答案C4.sin220°+cos250°+sin20°cos50°=()A.-1B.2C.43D.34解析原式=-12[cos(20°+20°)-cos(20°-20°)]+12[cos(50°+50°)+cos(50°-50°)]+12(sin70°-sin30°)=12(1-cos40°)+12(1+cos100°)+12(sin70°-sin30°)=1-12cos40°+12cos100°+12sin70°-12sin30°=34+12sin70°+12(cos100°-cos40°)=34+12sin70°-sin100°+40°2sin100°-40°2=34+12sin70°-sin30°sin70°=34.答案D5.cos(x+2020)-cos(x-2020)=.解析原式=-2sinx+2020+x-20202sinx+2020-\(x-2020\)2=-2sinxsin2020.答案-2sinxsin20206.cos37.5°cos22.5°=.解析cos37.5°cos22.5°=12(cos60°+cos15°)=14+12cos15°=2+√6+√28.3答案2+√6+√287.cos15°cos60°cos75°=.解析原式=12cos15°cos75°=14[cos90°+cos(-60°)]=18.答案188.求值:sin42°-cos12°+sin54°.解原式=sin42°-sin78°+sin54°=-2cos60°sin18°+sin54°=sin54°-sin18°=2cos36°sin18°=2×2cos36°sin18°cos18°2cos18°=2cos36°\(sin18°cos18°+sin18°cos18°\)2cos18°=2sin36°cos36°2cos18°=sin36°cos36°+sin36°cos36°2cos18°=sin72°2cos18°=12.9.求证:2cos20°+2sin20°-12cos20°-2sin20°-1·tan25°=cos15°sin15°.证明左边=2cos20°sin25°+2sin20°sin25°-sin25°2cos20°cos25°-2sin20°cos25°-cos25°=sin45°-sin\(-5°\)-cos45°+cos\(-5°\)-sin25°cos45°+cos\(-5°\)-sin45°-sin\(-5°\)-cos25°=sin5°+cos5°-sin25°sin5°+cos5°-cos25°=sin5°+sin85°-sin25°cos85°+cos5°-cos25°=sin5°+2cos55°sin30°-2s...
