Proof cos(x+y) cos(x-y)= cos2x + cos2y - 1 ?

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Answer 1 · 2017-03-13T13:28:37

$cos(x+y)cos(x-y)=1/2(cos2x+cos2y)$

not $cos2x+cos2y-1$

$cos(x+y)cos(x-y)$

= $(cosxcosy-sinxsiny)(cosxcosy+sinxsiny)$

= $cos^2xcos^2y-sin^2xsin^2y$

= $cos^2x(1-sin^2y)-(1-cos^2x)sin^2y$

= $cos^2x-cos^2xsin^2y-sin^2y+cos^2xsin^2y$

= $cos^2x-sin^2y$

= $1/2(2cos^2x)-1/2(2sin^2y)$

= $1/2(1+cos2x)-1/2(1-cos2y)$

= $1/2(cos2x+cos2y)$