Proof $cos(x+y) cos(x-y)= cos^2x + cos^2y - 1$ ?

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Answer 1 · 2017-03-13T02:58:37

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

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

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

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

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

$=cos^2x-cancel(cos^2xsin^2y)-sin^2y+cancel(cos^2xsin^2y)$

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

$=cos^2x+cos^2y-1=RHS$

Proved

Proof cos(x+y) cos(x-y)= cos^2x + cos^2y - 1cos(x+y) cos(x-y)= cos^2x + cos^2y - 1?