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Question

How do you verify the identity: cos(x)-cos(y)sin(x)+sin(y)+sin(x)-sin(y)cos(x)+cos(y)=0?


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Solution

Verify the given identity:

Given identity: cos(x)-cos(y)sin(x)+sin(y)+sin(x)-sin(y)cos(x)+cos(y)=0.

Consider the expression: cos(x)-cos(y)sin(x)+sin(y)+sin(x)-sin(y)cos(x)+cos(y)

=cos(x)-cos(y)cos(x)+cos(y)+sin(x)-sin(y)sin(x)+sin(y)sinx+sinycosx+cosy=cos2x-cos2y+sin2x-sin2ysinx+sinycosx+cosy(a+b)(a-b)=a2-b2=sin2x+cos2x-sin2y+cos2ysinx+sinycosx+cosy=1-1sinx+sinycosx+cosysin2θ+cos2θ=1=0sinx+sinycosx+cosy=0

Therefore, the given identity, cos(x)-cos(y)sin(x)+sin(y)+sin(x)-sin(y)cos(x)+cos(y)=0 is proved.


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