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Question
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Solution
\[\text{ LHS }= \sin\frac{x}{2}\sin\frac{7x}{2} + \sin\frac{3x}{2}\sin\frac{11x}{2}\]
\[ = \frac{1}{2}\left[ 2\sin\frac{x}{2}\sin\frac{7x}{2} + 2\sin\frac{3x}{2}\sin\frac{11x}{2} \right]\]
\[ = \frac{1}{2}\left[ \cos\left( \frac{7x}{2} - \frac{x}{2} \right) - \cos\left( \frac{7x}{2} + \frac{x}{2} \right) + \cos\left( \frac{11x}{2} - \frac{3x}{2} \right) - \cos\left( \frac{11x}{2} + \frac{3x}{2} \right) \right]\]
\[ = \frac{1}{2}\left[ \cos3x - \cos4x + \cos4x - \cos7x \right]\]
\[ = \frac{1}{2}\left[ \cos3x - \cos7x \right]\]
\[ = \frac{1}{2}\left[ - 2\sin\left( \frac{3x + 7x}{2} \right)\sin\left( \frac{3x - 7x}{2} \right) \right]\]
\[ = \frac{1}{2}\left[ - 2\sin\left( 5x \right)\sin\left( - 2x \right) \right]\]
\[ = \sin\left( 5x \right)\sin\left( 2x \right) =\text{ RHS }\]
Hence, LHS = RHS
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