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$2\sin ^{2}B+ 4\cos (A+B) \sin A \sin B+ \cos 2 (A+B)$

Option 1)
$\sin 2A$

Option 2)
$\cos 2B$

Option 3)
$\cos 2A$

Option 4)
$\sin 2B$

Answers (1)

P Plabita

As we learnt in

Double Angle Formula -

Double angle formula

- wherein

These are formulae for double angles.

$2\sin ^{2}B+ 4\cos \left ( A+B \right )\sin A\sin B+\cos 2\left ( A+B \right )$

$= 1 - \cos 2B+4\cos \left ( A+B \right )\sin A\sin B+2\cos ^{2}\left ( A+B \right )-1$

$= -\cos 2B+2\cos ^{2}\left ( A+B \right )+4\cos \left ( A+B \right )\sin A\sin B$

$= - \cos 2B+2\cos \left ( A+B \right )\left [ \cos \left ( A+B \right ) +2\sin A\sin B\right ]$

$= - \cos 2B+2\cos \left ( A+B \right )\left [ \cos \left ( A+B \right ) +\cos \left ( A-B \right ) - \cos \left ( A+B \right ) ]$

$= - \cos 2B+ \cos 2A+\cos 2B$

$= \cos 2A$

Option 1)

$\sin 2A$

This option is incorrect

Option 2)

$\cos 2B$

This option is incorrect

Option 3)

$\cos 2A$

This option is correct

Option 4)

$\sin 2B$

This option is incorrect

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