cos(x+y)*cos(x-y)=?

Question

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Answer 1

cos(x+y)=cosxcosy-sinx siny; cosxcosy+sinx siny.

So the given expression becomes: cos^2(x)cos^2(y)-sin^2(x)sin^2(y).

This is the same as: (1-sin^2(x))(1-sin^2(y))-sin^2(x)sin^2(y).

1-sin^2(x)-sin^2(y)+sin^2(x)sin^2(y)-sin^2(x)sin^2(y)=cos^2(x)-sin^2(y)=(1-sin^2(x))-(1-cos^2(y))=cos^2(y)-sin^2(x).