1.两个式子分别平方：(sinx+siny)^2 =(sinx)^2 + 2sinxsiny + (siny)^2=16/25
(cosx+cosy)^2 =(cosx)^2 + 2cosxcosy + (cosy)^2=9/25
两式相加：(sinx)^2 + (cosx)^2 + 2sinxsiny + 2cosxcosy + (siny)^2 + (cosy)^2=16/25 + 9/25
1 + 2sinxsiny + 2cosxcosy + 1 =1
2(sinxsiny + cosxcosy)=-1
cos(x-y)=-1/2
2.[sin50+cos40*(1+tan60*tan10)]/(cos20)^2
={sin50 + cos40*[1 + (sin60sin10/cos60cos10)]} / (cos20)^2
={sin50 + cos40*[(cos60cos10/cos60cos10) + (sin60sin10/cos60cos10)]} / (cos20)^2
={sin50 + cos40*[(cos60cos10+sin60sin10)/(cos60cos10)]} / (cos20)^2
={sin50 + cos40*[cos(60-10) / (cos60cos10)]} / (cos20)^2
=[sin50 + (cos40cos50)/(cos60cos10)] / (cos20)^2
={sin50 + [cos40cos(90-40) / (1/2)cos10]} / (cos20)^2
=[sin(90-40) / (cos20)^2] + [2cos40sin40 / cos10*(cos20)^2]
=[(cos40) / (1+cos40)/2] + [(sin80) / cos10*(cos20)^2]
=[2cos40 / (1+cos40)] + [sin(90-10) / cos10*(cos20)^2]
=[2cos40 / (1+cos40)] + {(cos10) / cos10*[(1+cos40)/2]}
=(2cos40 + 2)/(1+cos40)
=2
3.(2cos^2 a-1)/[2tan*(45-a)*sin^2(45+a)]
= (2cos^2 a-1)/{2tan*(45-a)*[sin^2(90-(45-a)]}
= (2cos^2 a-1)/[2tan*(45-a)*cos^2(45-a)]
=(cos2a)/[2sin(45-a)cos(45-a)]
=(cos2a)/[sin(90-2a)]
=(cos2a)/(cos2a)=1
4.cos(a+b)=cosacosb - sinasinb=1/3
cos(a-b)=cosacosb + sinasinb=1/6
两式相加：2cosacosb=1/2
两式相减：2sinasinb=-1/6
tana*tanb=(sinasinb)/(cosacosb)=-1/3
5.在三角形中：tanC=tan[π-(A+B)]=-tan(A+B)
根据正切的两角和公式,变形后得：tanA+tanB={[tan(A+B)]*(1-tanAtanB)}
展开：tanA+tanB=tan(A+B) - tan(A+B)tanAtanB
将tanC=-tan(A+B)代入上式：tanA+tanB = -tanC + tanCtanAtanB
整理后：tanA+tanB+tanC = tanAtanBtanC
在三角形中：A+B+C=π ,则：A/2 + B/2 + C/2 =π/2
tan(A/2)*tan(B/2) + tan(B/2)*tan(C/2) + tan(C/2)*tan(A/2)
=tan(A/2)[tan(B/2)+tan(C/2)] + tan(B/2)*tan(C/2)
={tan[π/2 - (B/2 + C/2)]}*{[tan(B/2 + C/2)][1 - tan(B/2)*tan(C/2)]} + tan(B/2)*tan(C/2)
=cot(B/2 + C/2)*[tan(B/2 + C/2)][1 - tan(B/2)*tan(C/2)] + tan(B/2)*tan(C/2)
=1[1 - tan(B/2)*tan(C/2)] + tan(B/2)*tan(C/2)
=1 - tan(B/2)tan(C/2) + tan(B/2)*tan(C/2)
=1 1.两个式子分别平方：(sinx+siny)^2 =(sinx)^2 + 2sinxsiny + (siny)^2=16/25
(cosx+cosy)^2 =(cosx)^2 + 2cosxcosy + (cosy)^2=9/25
两式相加：(sinx)^2 + (cosx)^2 + 2sinxsiny + 2cosxcosy + (siny)^2 + (cosy)^2=16/25 + 9/25
1 + 2sinxsiny + 2cosxcosy + 1 =1
2(sinxsiny + cosxcosy)=-1
cos(x-y)=-1/2
2.[sin50+cos40*(1+tan60*tan10)]/(cos20)^2
={sin50 + cos40*[1 + (sin60sin10/cos60cos10)]} / (cos20)^2
={sin50 + cos40*[(cos60cos10/cos60cos10) + (sin60sin10/cos60cos10)]} / (cos20)^2
={sin50 + cos40*[(cos60cos10+sin60sin10)/(cos60cos10)]} / (cos20)^2
={sin50 + cos40*[cos(60-10) / (cos60cos10)]} / (cos20)^2
=[sin50 + (cos40cos50)/(cos60cos10)] / (cos20)^2
={sin50 + [cos40cos(90-40) / (1/2)cos10]} / (cos20)^2
=[sin(90-40) / (cos20)^2] + [2cos40sin40 / cos10*(cos20)^2]
=[(cos40) / (1+cos40)/2] + [(sin80) / cos10*(cos20)^2]
=[2cos40 / (1+cos40)] + [sin(90-10) / cos10*(cos20)^2]
=[2cos40 / (1+cos40)] + {(cos10) / cos10*[(1+cos40)/2]}
=(2cos40 + 2)/(1+cos40)
=2
3.(2cos^2 a-1)/[2tan*(45-a)*sin^2(45+a)]
= (2cos^2 a-1)/{2tan*(45-a)*[sin^2(90-(45-a)]}
= (2cos^2 a-1)/[2tan*(45-a)*cos^2(45-a)]
=(cos2a)/[2sin(45-a)cos(45-a)]
=(cos2a)/[sin(90-2a)]
=(cos2a)/(cos2a)=1
4.cos(a+b)=cosacosb - sinasinb=1/3
cos(a-b)=cosacosb + sinasinb=1/6
两式相加：2cosacosb=1/2
两式相减：2sinasinb=-1/6
tana*tanb=(sinasinb)/(cosacosb)=-1/3
5.在三角形中：tanC=tan[π-(A+B)]=-tan(A+B)
根据正切的两角和公式,变形后得：tanA+tanB={[tan(A+B)]*(1-tanAtanB)}
展开：tanA+tanB=tan(A+B) - tan(A+B)tanAtanB
将tanC=-tan(A+B)代入上式：tanA+tanB = -tanC + tanCtanAtanB
整理后：tanA+tanB+tanC = tanAtanBtanC
在三角形中：A+B+C=π ,则：A/2 + B/2 + C/2 =π/2
tan(A/2)*tan(B/2) + tan(B/2)*tan(C/2) + tan(C/2)*tan(A/2)
=tan(A/2)[tan(B/2)+tan(C/2)] + tan(B/2)*tan(C/2)
={tan[π/2 - (B/2 + C/2)]}*{[tan(B/2 + C/2)][1 - tan(B/2)*tan(C/2)]} + tan(B/2)*tan(C/2)
=cot(B/2 + C/2)*[tan(B/2 + C/2)][1 - tan(B/2)*tan(C/2)] + tan(B/2)*tan(C/2)
=1[1 - tan(B/2)*tan(C/2)] + tan(B/2)*tan(C/2)
=1 - tan(B/2)tan(C/2) + tan(B/2)*tan(C/2)
=1