(3sin^2a/2-2sina/2cosa/2+cos^2a/2)/tana+cota

sin^2a=1/2（1-cos2a);cos^2a=1/2（1+cos2a);
1－tan2α
cos2α＝——————=-4/5 2/3sin^2a+1/4cos^2a=5/8
1＋tan2α

由3sin2a-2sin2b.故6sina*cosa=4sinb*cosb,
故3sina*根号(1-(sina)^2)=2sinb*根号(1-(sinb)^2),
故3*根号((sina)^2-(sina)^4)=2*根号((sinb)^2-(sinb)^4)
故9(sina)^2-9(sina)^4=4(sinb)^2-4(sinb)^4=2*(1-3(sina)^2)-[1-(sina)^2]^2
=2-6(sina)^2-1+6(sina)^2-9(sina)^4
故9(sina)^2=1,sina=1/3,
故2(sinb)^2=1-3*1/9=2/3,(sinb)^2=1/3,cos2b=1-2(sinb)^2=1/3,
故sin(a+2b)=sina*cos2b+cosa*sin2b=1/3*1/3+2*根号2/3*2*根号2/3=1,
故a+2b=派/2=a+2b=3.14/2

(3SINA-2COSA)/(2SINA+COSA)
分子分母同除以COSA
=(3 TANA-2)/(2 TANA+1)
=8/3.
3SIN^2A-4SINACOSA
=(3SIN^2A-4SINACOSA)/1
=(3SIN^2A-4SINACOSA)/( SIN^2A+COS^2A)
分子分母同除以COS^2A
=(3 TAN^2A-4 TANA)/( TAN^2A+1)
=4.

tan(a+Pai/4)=(tana+1)/(1-tana)=2
tana+1=2-2tana
tana=1/3
sina/cosa=1/3
cosa=3sina
(cosa)^2+(sina)^2=1
10sin^2a=1
sin^2a=1/10
cos2a+3sin^2a+tan2a
=cos^2a-sin^2a+3sin^2a+2tana/(1-tan^2a)
=cos^2a+2sin^2a+(2/3)/(1-1/9)
=1+sin^2a+(2/3)/(8/9)
=1+1/10+2/3*9/8
=11/10+3/4
=22/20+15/20
=37/20

因为3(sinA)^2=1-2(sinB)^2=cos2B
3sin2A/2=sin2B
(cos2B)^2+(sin2B)^2=1
所以[3(sinA)^2]^2+(3sin2A/2)^2=1
-->9(sinA)^4+9(sinAcosA)^2=1
-->9(sinA)^4+9(sinA)^2[1-(sinA)^2]=1
设(sinA)^2=t,则9t^2+9t(1-t)=1,
解得t=1/9,从而sinA=1/3,cosA=2*根号2/3.
因为2B∈(0,2π)且cos2B=3(sinA)^2=1/3>0,所以2B∈(0,π).
因为A∈(0,π/2),所以(π/2-A)∈(0,π/2).
由cos2B=sinA=cos(π/2-A)且余弦函数在(0,π/2)有单调性,
可得2B=π/2-A
即A+2B=π/2.证毕.

3(sinA)^2+2(sinB)^2=5sinA
(sinA)^2+(sinB)^2=5sinA/2-(sinA)^2/2
5sinA/2-(sinA)^2/2=-(1/2)(sinA-5/2)^2+25/4
sinA=-1 5sinA/2-(sinA)^2/2=-49/8+25/4=1/8
sinA=1 5sinA/2-(sinA)^2/2=-9/8+25/4=41/8
1/8 ≤ (sinA)^2+(sinB)^2 ≤ 41/8

sinA/cosA=tanA=2
sinA=2cosA
代入sin²A+cos²A=1
sin²A=4/5
cos²A=1/5
sinAcosA=2cosA*cosA=2cos²A=2/5
所以原式=12/5+6/5-2/5=16/5

3sin²a+2sin²b-2sina=2(sin²a+sin²b)+sin²a-2sina=0,则y=(2sina-sin²a)/2=(-(sina-1)²+1)/2,此时,易得当sina=1时,y取得最大值,y的最大值为1/2

因为由公式 cos^2a=1/1+tan^2a 可得cos^2a=1/5
原式=cos^2a(2/3tan^2a+1/4)
=1/5(8/3+1/4)
=7/12

tan(a+π/4)=2
tana=1/3
sin^2a=1/10
cos2a+3sin^2a+tan a
=1-2sin^2a+3sin^2a+tan a
=1+sin^2a+tan a
=1+1/10+1/3
=43/30

因为 2cos^2a+3cosasina-3sin^2a=1=cos^2a+sin^2a
所以 原式为
cos^2a+3cosasina-4sin^2a=0
(cosa+4sina)(cosa-sina)=0
因此 cosa=-4sina,即 tana= -1/4
cosa=sina,即 tana=1

题目有误,条件不足
已知a.b为锐角,且满足3sin^2a+2sin^2b=0.3sin(2A)-2sin(2B)=0求证a+2b=π/2
∵3sin^2A+2sin^2B=1
∴3sin^2A=cos(2B)
∵3sin(2A)-2sin(2B)=0
∴3sinAcosA=sin(2B)
把得到的两个式子相除,得
tanA=cot(2B)
∴tanA*tan(2B)=1
∵A、B为锐角,
∴A+2B=90°

tan(π+a)=3 a∈(0,2分之π)
所以
tana=3
sina/cosa=3
sina=3cosa
sin²a＋cos²a=1
10cos²a=1
cosa=√10/10
sina=3√10/10
3sin^2a+3sinacos^2a
=9/10＋3×3√10/10×1/10
=9/10＋9√10/100