证sin(x+y+z)+sin(x+y-z)+sin(x-y+z)+sin(x-y-z)=4sinxcosycosz应该是用和差化积或积化和差那些公式去做的.

Question

证sin(x+y+z)+sin(x+y-z)+sin(x-y+z)+sin(x-y-z)=4sinxcosycosz
应该是用和差化积或积化和差那些公式去做的.

Answer 1

sin(x+y+z)+sin(x+y-z)
=2sin(x+y)cosz
sin(x-y+z)+sin(x-y-z)
=2sin(x-z)cosz
所以左边=2sin(x+y)cosz+2sin(x-y)cosz
=2cosz[sin(x+y)+sin(x-y)]
=4coszsinxcosy=右边