Question

数学题，哪位大神帮帮忙，要过程，谢谢。

Answer 1

1+根号（3）tan 10= (cos10 + 根号（3）sin10)/cos10= 2(sin10cos30+ cos10sin30)/cos10 = 2sin40/cos10
原式分子= 2sin50sin40/cos10 – cos20 = 2sin50cos50/cos10 – cos20 = sin100/cos10 – cos20= 1-cos20
原式=根号（1-cos20）/cos 80= 根号（2）sin10/cos80= 根号（2）

1/tan5 – tan5 = cos5/sin5 – sin5/cos5 = (cos^2 5 – sin^2 5)/ (sin5cos5) =2 cos10/sin10
(1+cos20 )/2sin20 = sin^210/2sin10cos10 = ½ sin10/cos10
原式= ½ sin10/cos10 – 2 cos10 = ½

Answer 2

（1）解：∵1+√3tan10°=(cos10°+√3sin10°)/cos10°
=2(cos60°cos10°+sin60°sin10°)/cos10°=2cos50°/cos10°
∴分子=2cos50°sin50°/cos10°-cos20°=sin100°/cos10°-cos20°=sin80°/sin80°-cos20°
=1-cos20°=2(sin10°)^2；
又∵分母=cos80°√2(sin10°)^2=√2cos80°sin10°=√2(sin10°)^2；
∴原式=2(sin10°)^2/√2(sin10°)^2=√2。
（2）∵sin10°(cos5°/sin5°-sin5°/cos5°)=sin10°[(cos5°)^2-(sin5°)^2/cos5°sin5°]
=2sin10°cos10°/sin10°=2cos10°；
又 ∵（1+cos20°)/2sin20°=2(cos10°)^2/4sin10°cos10°=cos10°/2sin10°；
∴原式=cos10°/2sin10°-2cos10°=（cos10°-4cos10°sin10°)/2sin10°=(cos10°-2sin20°)/2sin10°
=[cos10°-2sin(30°-10°)]/2sin10°=√3sin10°/2sin10°=√3/2