Mathematics

0, 用和角公式爆開即可。

2014-09-07 05:38:47 補充：
(cosx + cosy + cosz) / cos(x + y + z) = (sinx + siny + sinz) / sin(x + y + z) (cosx + cosy + cosz) sin(x + y + z) = (sinx + siny + sinz) cos(x + y + z) ½ [sin(2x+y+z) + sin(y+z)] + ½ [sin(x+2y+z) + sin(z+x)] + ½ [sin(x+y+2z) + sin(x+y)]
= ½ [sin(2x+y+z) - sin(y+z)] + ½ [sin(x+2y+z) - sin(z+x)] + ½ [sin(x+y+2z) - sin(x+y)] ½ sin(y+z) + ½ sin(z+x) + ½ sin(x+y) = - ½ sin(y+z) - ½ sin(z+x) - ½ sin(x+y) sin(x+y) + sin(y+z) + sin(z+x) = 0
Alternative method by complex number : Let (cosx + cosy + cosz) / cos(x+y+z) = (sinx + siny + sinz) / sin(x+y+z) = k ,
then cosx + cosy + cosz = k cos(x+y+z) and isinx + isiny + isinz = k isin(x+y+z)
⇒ (cosx + isinx) + (cosy + isiny) + (cosz + isinz) = k( cos(x+y+z) + isin(x+y+z) )
cos(-y-z) + isin(-y-z) + cos(-x-z) + isin(-x-z) + cos(-x-y) + isin(-x-y) = k
cos(x+y) + cos(y+z) + cos(z+x) - i( sin(x+y) + sin(y+z) + sin(z+x) ) = k
k is real since x, y and z are real , so the coefficient of imaginary part
sin(x+y) + sin(y+z) + sin(z+x) = 0

2014-09-10 14:08:53 補充：
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