What is $(sin x cos y + cos x sin y) (sin x cos y - cos x sin y)$ equal to?

{cos}^{2} x - {cos}^{2} y

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{cos}^{2} x - {sin}^{2} y

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{sin}^{2} x - {cos}^{2} y

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{sin}^{2} x - {sin}^{2} y

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Solution

The correct option is D ${sin}^{2} x - {sin}^{2} y$
given, $(sin x cos y + cos x sin y) (sin x cos y - cos x sin y)$

$= {sin}^{2} x {cos}^{2} y - {cos}^{2} x {sin}^{2} y$

$= {sin}^{2} x {cos}^{2} y - (1 - {sin}^{2} x) {sin}^{2} y$

$= {sin}^{2} x {cos}^{2} y - {sin}^{2} y + {sin}^{2} x {sin}^{2} y$

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

$= {sin}^{2} x - {sin}^{2} y$ .

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