Question

Prove that:

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

Answer 1

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