Sum

Prove that:

\[2\sin\frac{5\pi}{12}\cos\frac{\pi}{12} = \frac{\sqrt{3} + 2}{2}\]

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#### Solution

\[LHS = 2\left( \sin \frac{5\pi}{12} \right) \left( \cos \frac{\pi}{12} \right)\]

\[ = \sin \left( \frac{5\pi}{12} + \frac{\pi}{12} \right) + \sin \left( \frac{5\pi}{12} - \frac{\pi}{12} \right) \left[ \because 2 \sin A \cos B = \sin (A + B) + \sin (A - B) \right]\]

\[ = \sin \frac{\pi}{2} + \sin \frac{\pi}{3}\]

\[ = 1 + \frac{\sqrt{3}}{2}\]

\[ = \frac{2 + \sqrt{3}}{2}\]

\[RHS = \frac{2 + \sqrt{3}}{2}\]

Hence, LHS = RHS

Concept: Transformation Formulae

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