$sin (α + β) cos (α - β) + cos (α + β) sin (α - β) =$

sin 2 β

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cos 2 α

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cos 2 β

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sin 2 α

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Solution

The correct option is D $sin 2 α$
$sin (α + β) cos (α - β) + cos (α + β) sin (α - β)$

As we know that,

$sin (A + B) = sin A cos B + cos A sin B$

Here,

$A = α + β$

$B = α - β$

Therefore,

$sin (α + β) cos (α - β) + cos (α + β) sin (α - β)$

$= sin ((α + β) + (α - β))$

$= sin (α + β + α - β)$

$= sin (α + α)$

$= sin 2 α$

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