Question

如何证明cos a +cos b =2*(cos(a+b)/2)*(cos(a-b)/2) 速度

Answer 1

可以从右往左证明：

右边 = 2*(cos(a+b)/2)*(cos(a-b)/2)

= 2 * cos(a/2+b/2) * (cos(a/2-b/2)

= 2 * { cosa/2 cosb/2 - sina/2 sinb/2 ) * { cosa/2 cosb/2 + sina/2 sinb/2 )

= 2 * { (cosa/2)^2 (cosb/2)^2 - [(1-(cosa/2)^2][1- (cosb/2 ) ^2 }

= 2 * { (cosa/2)^2 (cosb/2)^2 - [1 -(cosa/2)^2- (cosb/2 ) ^2+(cosa/2)^2 (cosb/2)^2 ] }

= 2 * { (cosa/2)^2 (cosb/2)^2 - 1 + (cosa/2)^2+ (cosb/2 ) ^2 - (cosa/2)^2 (cosb/2)^2 }

= 2 * { - 1 + (cosa/2)^2 + (cosb/2 ) ^2 }

= -2 + 2 (cosa/2)^2 + 2 (cosb/2 ) ^2

= 2 (cosa/2)^2 -1 + 2 (cosb/2 ) ^2 - 1

= cos a +cos b = 左边,得证.