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如何证明cos a +cos b =2*(cos(a+b)/2)*(cos(a-b)/2) 速度
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如何证明cos a +cos b =2*(cos(a+b)/2)*(cos(a-b)/2) 速度
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答案和解析
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.
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.
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