Question

Prove that:

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

 

Answer 1

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