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Question
\[2 \left( 1 - 2 \sin^2 7x \right) \sin 3x\] is equal to
Options
\[\sin 17x - \sin 11x\]
\[\sin 11x - \sin 17x\]
\[\cos 17x - \cos 11x\]
\[\cos 17x + \cos 11x\]
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Solution
\[\sin 17x - \sin 11x\]
\[\text{ We have } , \]
\[ 2\left( 1 - 2 \sin^2 7x \right) \sin 3x = 2\left( \cos 14x \right) \sin3x \]
\[ \left[ \because \cos2x = 1 - 2 \sin^2 x \right]\]
\[ = 2 \sin3x \cos 14x\]
\[ = \sin 17x - \sin 11x\]
\[ \left[ \because 2 sinA cosB = \sin\left( A + B \right) - \sin\left( A - B \right) \right] \]
\[ \therefore 2\left( 1 - 2 \sin^2 7x \right) \sin 3x = \sin 17x - \sin 11x\]
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