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Prove That: 2 Sin 5 π 12 Sin π 12 = 1 2

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Question

Prove that:

\[2\sin\frac{5\pi}{12}\sin\frac{\pi}{12} = \frac{1}{2}\]

Sum

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Solution

\[LHS = 2\left( \sin \frac{5\pi}{12} \right) \left( \sin \frac{\pi}{12} \right)\]
\[ = \cos \left( \frac{5\pi}{12} - \frac{\pi}{12} \right) - \cos \left( \frac{5\pi}{12} + \frac{\pi}{12} \right) \left[ \because 2 \sin A \sin B = \cos (A - B) - \cos (A + B) \right]\]
\[ = \cos \frac{\pi}{3} - \cos \frac{\pi}{2}\]
\[ = \frac{1}{2} - 0\]
\[ = \frac{1}{2}\]
\[RHS = \frac{1}{2}\]
Hence, LHS = RHS

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Transformation Formulae

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Q 2.1 Q 1.4 Q 2.2

Chapter 8: Transformation formulae - Exercise 8.1 [Page 6]

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R.D. Sharma Mathematics [English] Class 11

Chapter 8 Transformation formulae
Exercise 8.1 | Q 2.1 | Page 6

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