求证(sina+sinb)^2+(cosa+cosb)^2=4cos^2[(a-b)/2]

求证(sina+sinb)^2+(cosa+cosb)^2=4cos^2[(a-b)/2]
求证(sina+sinb)^2+(cosa+cosb)^2=4cos^2[(a-b)/2]

求证(sina+sinb)^2+(cosa+cosb)^2=4cos^2[(a-b)/2]
证明：
左边=(sina)^2+(sinb)^2+2sinasinb+(cosa)^2+(cosb)^2+2cosacosb
=2+2(cosacosb+sinasinb)
=2+2cos(a-b)
=2+2{2cos^2[(a-b)/2]-1}
=4cos^2[(a-b)/2]
=右边
证毕

左边=(sin²a+cos²a)+(sin²b+cos²b)+2(cosacosb+sinasinb)
=1+1+2cos(a-b)
=2+2{2cos²[(a-b)/2]-1}
=4cos²[(a-b)/2]=右边
命题得证