Question

cos40°(1+根号3tan10°)

Answer 1

【1】单位：度“º”。【2】tan50=tan(60-10)=(tan60-tan10)/(1+tan60tan10)=(tan60-tan10)/[1+(√3)tan10].===>1+(√3)tan10=(tan60-tan10)/tan50=(tan60-tan10)cos50/sin50=(tan60-tan10)cos50/cos40.===>[1+(√3)tan10]cos40=(tan60-tan10)cos50,【3】tan60-tan10=(sin60/cos60)-(sin10/cos10)=[sin60cos10-cos60sin10]/(cos60cos10)=sin(60-10)/[cos60cos10]=sin50/(cos60cos10)=(2sin50cos50)/(2cos50cos60cos10)=sin100/(2cos50cos60cos10)=cos10/(2cos50cos60cos10)=1/(2cos50cos60).===>(tan60-tan10)cos50=1/(2cos60)=1.∴原式=1.

Answer 2

cos40°(1+√3tan10°)
=cos40°(1+√3sin10°/cos10°)
=cos40°(cos10°+√3sin10°)/cos10°
=2cos40°(cos10°/2+√3sin10°/2)/cos10°
=2cos40°(sin30°cos10°+cos30°sin10°)/cos10°
=2cos40°sin40°/cos10°
=sin80°/cos10°
=1