sin(x+y+z)
=sinxcosycosz+cosxsinycosz+cosxcosysinz-sinxsinysinz
=cosxcosycosz(tanx+tany+tanz-tanxtanytanz)
cos(x+y+z)
=cosxcosycosz-cosxsinysinz-sinxcosysinz-sinxsinycosz
=cosxcosycosz(1-tanxtany-tanytanz-tanztanx)
tan(x+y+z)
=[tanx+tany+tanz-tanxtanytanz]/[1-tanxtany-tanytanz-tanztanx]
cot(x+y+z)
=[cotxcotycotz-cotx-coty-cotz]/[cotxcoty+cotycotz+cotzcotx-1]
已知sin(x+y+z)
=sinxcosycosz+cosxsinycosz+cosxcosysinz-sinxsinysinz
所以
sin(x+y-z)
=sinxcosycosz+cosxsinycosz-cosxcosysinz+sinxsinysinz
sin(x-y+z)
=sinxcosycosz-cosxsinycosz+cosxcosysinz+sinxsinysinz
sin(-x+y+z)
=-sinxcosycosz+cosxsinycosz+cosxcosysinz+sinxsinysinz
sin(x+y+z)
=sinxcosycosz+cosxsinycosz+cosxcosysinz-sinxsinysinz
所以4sinxsinysinz =sin(x+y-z) +sin(x-y+z) +sin(-x+y+z) -sin(x+y+z)
sin(x-y-z)
=sinxcosycosz-cosxsinycosz-cosxcosysinz-sinxsinysinz
sin(-x-y+z)
=-sinxcosycosz-cosxsinycosz+cosxcosysinz-sinxsinysinz
sin(-x+y+z)
=-sinxcosycosz+cosxsinycosz+cosxcosysinz+sinxsinysinz
sin(x+y+z)
=sinxcosycosz+cosxsinycosz+cosxcosysinz-sinxsinysinz
全相加
2cosxcosysinz-2sinxsinysinz =sin(x-y-z) +sin(-x-y+z) +sin(-x+y+z) +sin(x+y+z)
再利用4sinxsinysinz =sin(x+y-z) +sin(x-y+z) +sin(-x+y+z) -sin(x+y+z)
得到4cosxcosysinz= +sin(-x-y+z) +sin(-x+y+z) +sin(x+y-z) +sin(x-y+z)
其他自己仿照这2个,推理吧!
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