Question

sin40°（tan10°-根号3）

1.sin40°（tan10°-根号3）
2. sin10°分之1-cos10°分之根号3
3. tan70°cos10°（根号3tan20°-1）
4. sin50°（1+根号3tan10°）
5. sin50°（1+根号3tan10°）

Answer 1

①sin40°(tan10°-根号3)
=sin40*sin10/cos10-根号3*sin40
=sin40*cos80/cos10-根号3*sin40
=sin40*cos80/2sin40*cos40-根号3*sin40
=(1/2)*cos80/cos40-根号3*sin40
=[(1/2)*cos80-根号3sin40*sin40]/cos40
=[(1/2)cos80-根号3/2*sin80]/cos40
=(sin30*cos80-cos30*sin80)/cos40
=sin(30-80)/cos40
=-sin50/cos40
=-1
②
1/sin10-√3/cos10
=(cos10-√3sin10)/(sin10cos10)
=2[(cos10)/2-(√3sin10)/2]/(sin20)/2
=4(sin30cos10-cos30sin10)/sin20
=4sin20/sin20
=4
③
tan70°*cos10°*(√3 *tan20° -1)
=tan70°*cos10°*(tan60 °*tan20° -1)
=tan70°*cos10°*[(sin60 °*sin20°/cos60°*cos20°)-1]
=tan70°*cos10°*(sin60 °*sin20°-cos60°*cos20°)/(cos60 °*cos20°)
=tan70°*cos10°*[-cos(60°+20°)]/(cos60 °*cos20°)
=-tan70°*cos10°*cos80°/(cos60 °*cos20°)
=-tan70°*cos10°*sin10°/(cos60 °*cos20°)
=-(sin70°/cos70°)*(1/2)*sin20°/(cos60 °*cos20°)
=-(cos20°/sin20°)*sin20°/(2cos60 °*cos20°)
=-1/(2cos60 °)
=-1/4
④
sin50（1+√3tan10）
=sin50[1+√3(sin10/cos10)]
=sin50[(cos10+√3sin10)/cos10]
=sin50[2sin(30+10)/cos10]
=(2sin50sin40)/cos10°
=(2cos40°*sin40°)/cos10°
=sin80°/cos10°
=cos10°/cos10°
=1

Answer 2

1. sin40°(sin10°/cos10°-sin60°/cos60°) √3=tan60°=sin60°/cos60°
=sin40°*(sin10°cos60°-cos10°sin60°)/cos10°*cos60°
=-sin40°*sin50°/(cos10°*1/2)
=-2sin40°cos40°/cos10°
=-sin80°/cos10°=-1
2.1/sin10°-sin60°/cos10°cos60°
=(cos10°cos60°-sin10°sin60°)/(sin10°cos10°cos60°)
=cos70°/(sin10°cos10°cos60°)
=2cos70°/(sin20°*1/2)
=4cos70°/sin20°=4sin20°/sin20°=4
3. tan70°cos10°（根号3tan20°-1）
= tan70°cos10°[(sin60°sin20°/cos60°cos20°)-1]
=tan70°cos10°[(sin60°sin20°-cos60°cos20°)/cos60°cos20°]
=-tan70°cos10°*cos80°/cos60°cos20°
=-tan70°cos10°*sin20°/(1/2*cos20°)
=-tan70°2cos10°*sin20°/cos20°)
=-tan70°*tan20°
=-1
4.sin50°（1+根号3tan10°）
=sin50°(1+sin60°sin10°/cos60°cos10°)
=sin50°(sin60°sin10°+cos60°cos10°)/cos60°cos10°
=sin50°cos50°/cos60°cos10°
=2sin50°cos50°/cos10°
=sin100°/cos10°
=1