Question

Prove that: 

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

Answer 1

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