Tính giá trị của các biểu thức sau:

a) $A = \frac{{\sin \frac{\pi }{{15}}\cos \frac{\pi }{{10}} + \sin \frac{\pi }{{10}}\cos \frac{\pi }{{15}}}}{{\cos \frac{{2\pi }}{{15}}\cos \frac{\pi }{5} - \sin \frac{{2\pi }}{{15}}\sin \frac{\pi }{5}}}$;

b) $B = \sin \frac{\pi }{{32}}\cos \frac{\pi }{{32}}\cos \frac{\pi }{{16}}\cos \frac{\pi }{8}$.

Lời giải:

a) Ta có:

$A = \frac{{\sin \frac{\pi }{{15}}\cos \frac{\pi }{{10}} + \sin \frac{\pi }{{10}}\cos \frac{\pi }{{15}}}}{{\cos \frac{{2\pi }}{{15}}\cos \frac{\pi }{5} - \sin \frac{{2\pi }}{{15}}\sin \frac{\pi }{5}}}$$= \frac{{\sin \frac{\pi }{{15}}\cos \frac{\pi }{{10}} + \cos \frac{\pi }{{15}}\sin \frac{\pi }{{10}}}}{{\cos \frac{{2\pi }}{{15}}\cos \frac{\pi }{5} - \sin \frac{{2\pi }}{{15}}\sin \frac{\pi }{5}}}$

$= \frac{{\sin \left( {\frac{\pi }{{15}} + \frac{\pi }{{10}}} \right)}}{{\cos \left( {\frac{{2\pi }}{{15}} + \frac{\pi }{5}} \right)}}$$= \frac{{\sin \frac{\pi }{6}}}{{\cos \frac{\pi }{3}}} = \frac{{\frac{1}{2}}}{{\frac{1}{2}}} = 1$.

b) Ta có:

$B = \sin \frac{\pi }{{32}}\cos \frac{\pi }{{32}}\cos \frac{\pi }{{16}}\cos \frac{\pi }{8}$$= \left( {\frac{1}{2}.2\sin \frac{\pi }{{32}}\cos \frac{\pi }{{32}}} \right)\cos \frac{\pi }{{16}}\cos \frac{\pi }{8}$

$= \frac{1}{2}\sin \left( {2.\frac{\pi }{{32}}} \right)\cos \frac{\pi }{{16}}\cos \frac{\pi }{8}$$= \frac{1}{2}\sin \frac{\pi }{{16}}\cos \frac{\pi }{{16}}\cos \frac{\pi }{8}$

$= \frac{1}{4}.2\sin \frac{\pi }{{16}}\cos \frac{\pi }{{16}}\cos \frac{\pi }{8}$$= \frac{1}{4}\sin \frac{\pi }{8}\cos \frac{\pi }{8} = \frac{1}{8}.2\sin \frac{\pi }{8}\cos \frac{\pi }{8}$

$= \frac{1}{8}\sin \frac{\pi }{4} = \frac{1}{8}.\frac{{\sqrt 2 }}{2} = \frac{{\sqrt 2 }}{{16}}$.