How does one prove sin(x+y) - sin(x-y) = 2cos(x) sin(y)?

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How does one prove sin(x+y) - sin(x-y) = 2cos(x) sin(y)?

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Answer 1 · 2018-04-16T04:02:58

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Explanation:

$sin (x + y) - sin (x - y)$ $sin (x+y)-sin (x-y)$
$= sin x cos y + sin y cos x - (sin x cos y - sin y cos x)$ $=sinxcosy+sinycosx-(sinxcosy-sinycosx)$
$= sin x cos y + sin y cos x - sin x cos y + sin y cos x$ $=sinxcosy+sinycosx-sinxcosy+sinycosx$
$=cancel (sinxcosy)+sinycosx-cancel (sinxcosy)+sinycosx$
$=sinycosx+sinycosx$
$=2sinycosx$ henceforth proved.

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How does one prove sin(x+y) - sin(x-y) = 2cos(x) sin(y)?

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