Question #10fa9

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Question #10fa9

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Answer 1 · 2017-11-14T18:40:05

$\frac{cos (x + y) + cos (x - y)}{cos x sin y} = 2 cot y$ $(cos(x+y)+cos(x-y))/(cosxsiny)=2coty$

Apply the identity $cos (α \pm β) = cos α cos β \pm (- sin α sin β)$ $cos(alpha+-beta)=cosalphacosbeta+-(-sinalphasinbeta)$

$\frac{cos x cos y - sin x sin y + cos x cos y + sin x sin y}{cos x sin y}$ $(cosxcosy-sinxsiny+cosxcosy+sinxsiny)/(cosxsiny)$
$\frac{2 cos x cos y}{cos x sin y}$ $(2cosxcosy)/(cosxsiny)$
$\frac{2 cos y}{sin y}$ $(2cosy)/siny$

Definitionally, this is $2 cot y$ $2coty$

Answer 2 · 2017-11-14T18:53:18

$see explanation$ $"see explanation"$

Explanation:

$using the addition formulae for cos$ $"using the "color(blue)"addition formulae for cos"$

$∙ x cos (A \pm B) = cos A cos B \mp sin A sin B$ $•color(white)(x)cos(A+-B)=cosAcosB∓sinAsinB$

$∙ x cot y = \frac{cos y}{sin y}$ $•color(white)(x)coty=cosy/siny$

$consider the LHS$ $"consider the LHS"$

$\frac{cos (x + y) + cos (x - y)}{cos x sin y}$ $(cos(x+y)+cos(x-y))/(cosxsiny)$

$=(cosxcosycancel(-sinxsiny)+cosxcosycancel(+sinxsiny))/(cosxsiny)$

$=(2cosxcosy)/(cosxsiny)$

$=(2cancel(cosx)cosy)/(cancel(cosx)siny)$

$=2cosy/siny$

$=2coty="RHS"rArr" proved"$

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Question #10fa9

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