急解!已知sina+sinb=根号2 cosa+cosb=2根号3/3. 求cos（a-b）,tan（a+b)/2,sin(a+b),tana*tanb的值

Question

急解!已知sina+sinb=根号2 cosa+cosb=2根号3/3. 求cos（a-b）,tan（a+b)/2,sin(a+b),tana*tanb的值

Answer 1

(sina+sinb)²=2 ,(cosa+cosb)²=4/3,前后两式相加得：2+2(cosacosb+sinasinb)=10/3,cos(a-b)=2/3；后式减前式得：2(cosacosb-sinasinb)+cos2a+cos2b=-2/3,2cos(a+b)+2cos(a+b)cos(a-b)=-2/3,cos(a+b)=-1/5,cos(a-b)+cos(a+b)=2/3-1/5,cosacosb=7/30,cos(a-b)-cos(a+b)=2/3+1/5,sinasinb=13/30,tana*tanb=13/7；sina+sinb=√2,cosa+cosb=2√3/3两式相乘得：sinacosa+sinbcosb+sin(a+b)=2√6/3,sin(a+b)cos(a-b)+sin(a+b)=2√6/3,sin(a+b)=2√6/5,tan（a+b)/2=[1-cos(a+b)]/sin(a+b)=(1+1/5)/(2√6/5)=√6/2.