How do you simplify $sin θ {cos}^{2} θ - sin θ$ $sin theta cos ^2 theta - sin theta$ ?

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How do you simplify $sin θ {cos}^{2} θ - sin θ$ $sin theta cos ^2 theta - sin theta$ ?

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Answer 1 · 2016-06-27T18:52:17

$- {sin}^{2} (θ)$ $-sin^2(theta)$

Explanation:

Remember that
$XXX {sin}^{2} (θ) + {cos}^{2} (θ) = 1$ $color(white)("XXX")sin^2(theta)+cos^2(theta)=1$
or (in a modified form)
$XXX {cos}^{2} (θ) - 1 = - {sin}^{2} (θ)$ $color(white)("XXX")color(blue)(cos^2(theta)-1=-sin^2(theta))$

Given the expression:
$XXX sin (θ) {cos}^{2} (θ) - sin (θ)$ $color(white)("XXX")sin(theta)cos^2(theta)-sin(theta)$
we can factor this as
$XXX sin (θ) \cdot ({cos}^{2} (θ) - 1)$ $color(white)("XXX")color(red)(sin(theta))*(color(blue)(cos^2(theta)-1))$

The using the earlier identity, we have
$XXX sin (θ) \cdot (- sin (θ))$ $color(white)("XXX")color(red)(sin(theta))*(color(blue)(-sin(theta)))$
or
$XXX - {sin}^{2} (θ)$ $color(white)("XXX")-sin^2(theta)$

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How do you simplify $sin θ {cos}^{2} θ - sin θ$ $sin theta cos ^2 theta - sin theta$ ?

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