Cho \(cos2x = \frac{1}{4}\). Tính: \(A = \cos \left( {x + \frac{\pi }{6}} \right)\cos \left( {x - \frac{\pi }{6}} \right)\); \(B = \sin \left( {x + \frac{\pi }{3}} \right)\sin \left( {x - \frac{\pi }{3}} \right)\).
Ta có:
\[A = \cos \left( {x + \frac{\pi }{6}} \right)\cos \left( {x - \frac{\pi }{6}} \right)\]
\[ = \frac{1}{2}\left[ {\cos \left( {x + \frac{\pi }{6} + x - \frac{\pi }{6}} \right) + \cos \left( {x + \frac{\pi }{6} - x + \frac{\pi }{6}} \right)} \right]\]
\[ = \frac{1}{2}\left[ {\cos 2x + \cos \frac{\pi }{3}} \right]\]
\[ = \frac{1}{2}\left[ {\frac{1}{4} + \frac{1}{2}} \right] = \frac{3}{8}\].
\(B = \sin \left( {x + \frac{\pi }{3}} \right)\sin \left( {x - \frac{\pi }{3}} \right)\)
\[ = - \frac{1}{2}\left[ {\cos \left( {x + \frac{\pi }{3} + x - \frac{\pi }{3}} \right) - \cos \left( {x + \frac{\pi }{3} - x + \frac{\pi }{3}} \right)} \right]\]
\[ = - \frac{1}{2}\left[ {\cos 2x - \cos \frac{{2\pi }}{3}} \right]\]
\[ = - \frac{1}{2}\left[ {\frac{1}{4} - \left( { - \frac{1}{2}} \right)} \right] = - \frac{3}{8}\].
Vậy \(A = \frac{3}{8},B = - \frac{3}{8}\).