Question #32396

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Question #32396

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Answer 1 · 2016-05-18T11:09:09

The equation as given is invalid.
See below for a proof that assumes the exponents were omitted and the equation should be:
$sin (x) - sin (x) {cos}^{2} (x) = {sin}^{3} (x)$ $sin(x)-sin(x)cos^2(x)=sin^3(x)$

Explanation:

$sin (x) - sin (x) {cos}^{2} (x)$ $sin(x)-sin(x)cos^2(x)$

factoring:
$XXX = sin (x) \cdot (1 - {cos}^{2} (x))$ $color(white)("XXX")=sin(x)*(1-cos^2(x))$

Since ${sin}^{2} (x) + {cos}^{2} (x) = 1$ $sin^2(x)+cos^2(x)=1$
$XXX$ $color(white)("XXX")$ we know this as a standard identity
$XXX$ $color(white)("XXX")$ or we could develop it easily from Pythagorean theorem and definition of $sin$ $sin$ and $cos$ $cos$

$sin (x) \cdot (1 - {cos}^{2} (x))$ $sin(x)*(color(red)(1-cos^2(x)))$
$XXX = sin (x) \cdot {sin}^{2} (x)$ $color(white)("XXX")=sin(x)*color(red)(sin^2(x))$

$XXX = {sin}^{3} (x)$ $color(white)("XXX")=sin^3(x)$

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Question #32396

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