1.已知tana=1/3,则COS^a+1/2sin2a=?

1.已知tana=1/3,则cos²a+1/2sin2a=?
因为sin²a+cos²a=1,sin2a=2sinacosa,sina/cosa=tana
故：cos²a+1/2sin2a
= cos²a+ sinacosa
=（cos²a+ sinacosa）/（sin²a+cos²a）（分子、分母同时除以cos²a）
=(1+tana)/(tan²a+1)
=(1+1/3)/(1/9+1)
=6/5
2.1/sin10°-√3/cos10°
=(cos10°-√3 sin10°)/( sin10°cos10°)
=4(1/2cos10°-√3/2sin10°)/(2sin10°cos10°)
=4(sin30°cos10°-cos30°sin10°)/ sin20°
=4sin(30°-10°)/sin20°
=4
3.使得cosx-cosy=a²-1/2,且sin[(x+y)/2]sin[(x-y)/2]=-a-5/4的实数a等于好多
因为cosx-cosy
=cos[(x+y)/2+(x-y)/2]-cos[(x+y)/2-(x-y)/2]
=cos[(x+y)/2]cos[(x-y)/2]-sin[(x+y)/2]sin[(x-y)/2]-cos[(x+y)/2]cos[(x-y)/2]-sin[(x+y)/2]sin[(x-y)/2]
=-2 sin[(x+y)/2]sin[(x-y)/2]= a²-1/2
故：sin[(x+y)/2]sin[(x-y)/2]= -a²/2+1/4=-a-5/4
故：a=3或a=-1
又：-2≤cosx-cosy≤2
而a=3时,cosx-cosy=a²-1/2=17/2＞2,故：舍去
故：a=-1
4.sin6°sin42°sin66°sin78°
= sin6°cos48°cos24°cos12°
= sin6°cos6°cos48°cos24°cos12°/ cos6°
=16 sin6°cos6°cos12°cos24°cos48°/ (16cos6°)
=8sin12°cos12°cos24°cos48°/ (16cos6°)
=4 sin24°cos24°cos48°/ (16cos6°)
=2sin48°cos48°/ (16cos6°)
=sin96°/ (16cos6°)
= cos6°/ (16cos6°)
=1/16
5.sin(π/4-X)=5/13 求cos2X/cos(π/4+X)的值
sin(π/4-X)=cos[π/2-(π/4-X)]= cos(π/4+X)= 5/13
sin(π/4-X)=sin(π/4)cosX-cos(π/4)sinX=5/13
即：√2/2•cosX-√2/2•sinX=5/13
故：cosX-sinX=5√2/13
两边平方：cos²X-2sinX cosX+sin²X =（5√2/13）²
即：1-sin(2X)=50/169
sin(2X)=119/169
故：cos2X=±√[1-sin² (2X)]= ±120/169
故：cos2X/cos(π/4+X)= ±120/169÷5/13=±24/13