How do you simplify $sin (u - v) cos v + cos (u - v) sin v$ $sin(u-v)cosv+cos(u-v)sinv$ ?

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How do you simplify $sin (u - v) cos v + cos (u - v) sin v$ $sin(u-v)cosv+cos(u-v)sinv$ ?

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Answer 1 · 2018-06-05T15:10:02

$sin u$ $sinu$

Explanation:

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

$∙ x sin (x \pm y) = sin x cos y \pm cos x sin y$ $•color(white)(x)sin(x+-y)=sinxcosy+-cosxsiny$

$∙ x cos (x \pm y) = cos x cos y \mp sin x sin y$ $•color(white)(x)cos(x+-y)=cosxcosy∓sinxsiny$

$∙ x {sin}^{2} x + {cos}^{2} x = 1$ $•color(white)(x)sin^2x+cos^2x=1$

$sin (u - v) cos v \leftarrow first term$ $sin(u-v)cosvlarrcolor(red)"first term"$

$= (sin u cos v - cos u sin v) cos v$ $=(sinucosv-cosusinv)cosv$

$=sinucos^2vcancel(-cosucosvsinv)$

$cos(u-v)sinvlarrcolor(red)"second term"$

$=(cosucosv+sinusinv)sinv$

$=cancel(cosucosvsinv)+sinusin^2v$

$"adding the 2 expansions gives"$

$sinucos^2v+sinusin^2v$

$=sinu(cos^2v+sin^2v)=sinu$

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How do you simplify $sin (u - v) cos v + cos (u - v) sin v$ $sin(u-v)cosv+cos(u-v)sinv$ ?

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Explanation:

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