Verify this is an identity? cos2x+cos2y/sinx+cosy=2cosy-2sinx

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Answer 1 · 2018-04-03T09:30:22

Please see below.

We know that

$(1) cos 2 θ = 2 {cos}^{2} θ - 1, or$ $color(red)((1)cos2theta=2cos^2theta-1,or$

$(2) cos 2 θ = 1 - 2 {sin}^{2} θ$ $color(red)((2)cos2theta=1-2sin^2theta$

We have ,

$\frac{cos 2 x + cos 2 y}{sin x + cos y} = 2 cos y - 2 sin x$ $(cos2x+cos2y)/(sinx+cosy)=2cosy-2sinx$

$LHS=(cos2x+cos2y)/(sinx+cosy)...to Apply (1)and (2)$

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

$=(2cos^2y-2sin^2x)/(sinx+cosy)$

$=(2(cos^2y-sin^2x))/(sinx+cosy)$

$=(2(cosy+sinx)(cosy-sinx))/((cosy+sinx))$

$=2(cosy-sinx)$

$=2cosy-2sinx$

$=RHS$