Proving Trigonometric Identities?

Question

$cos(x-y)sinx-sin(x-y)cosx=siny$

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Answer 1 · 2018-06-21T20:41:27

See below

Use the following identities:

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

The expression becomes

$(sin(x) sin(y) + cos(x) cos(y))sin(x) - (sin(x) cos(y) - cos(x) sin(y))cos(x)$

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

$= sin^2(x) sin(y) + sin(y) cos^2(x) = sin(y)(sin^2(x)+cos^2(x))$

And since $sin^2(x)+cos^2(x)=1$ , the result is proven