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

b) \(B = \sin \frac{\pi }{{32}}\cos \frac{\pi }{{32}}\cos \frac{\pi }{{16}}\cos \frac{\pi }{8} = \frac{1}{2}\sin \frac{\pi }{{16}}.\cos \frac{\pi }{{16}}.\cos \frac{\pi }{8} = \frac{1}{4}\sin \frac{\pi }{8}.\cos \frac{\pi }{8} = \frac{1}{8}\sin \frac{\pi }{4} = \frac{1}{8}.\frac{{\sqrt 2 }}{2} = \frac{{\sqrt 2 }}{{16}}\;.\)

\(B = \left( {\cos \frac{\pi }{9} + \cos \frac{{5\pi }}{9}} \right) + \cos \frac{{11\pi }}{9} = \left( {2\cos \frac{{\frac{\pi }{9} + \frac{{5\pi }}{9}}}{2}\cos \frac{{\frac{\pi }{9} - \frac{{5\pi }}{9}}}{2}} \right) + \cos \frac{{11\pi }}{9} = 2\cos \frac{\pi }{3}\cos \frac{{2\pi }}{9} + \cos \frac{{11\pi }}{9}\)

\( = \cos \frac{{2\pi }}{9} + \cos \frac{{11\pi }}{9} = 2\cos \frac{{\frac{{2\pi }}{9} + \frac{{11\pi }}{9}}}{2}\cos \frac{{\frac{{2\pi }}{9} - \frac{{11\pi }}{9}}}{2} = 2\cos \frac{{13\pi }}{{18}}\cos \frac{\pi }{2} = 0\)

Ta có:

\({\cos ^2}a + {\sin ^2}a = 1 \Rightarrow \sin a =  \pm \frac{4}{5}\)

Do \(0 < a < \frac{\pi }{2} \Leftrightarrow \sin a = \frac{4}{5}\)

\(\tan a = \frac{{\sin a}}{{\cos a}} = \frac{4}{3}\)

Ta có;

\(\begin{array}{l}\sin \left( {a + \frac{\pi }{6}} \right) = \sin a.\cos \frac{\pi }{6} + \cos a.\sin \frac{\pi }{6} = \frac{4}{5}.\frac{{\sqrt 3 }}{2} + \frac{3}{5}.\frac{1}{2} = \frac{{3 + 4\sqrt 3 }}{{10}}\\\cos \left( {a - \frac{\pi }{3}} \right) = \cos a.\cos \frac{\pi }{3} + \sin a.\sin \frac{\pi }{3} = \frac{3}{5}.\frac{1}{2} + \frac{4}{5}.\frac{{\sqrt 3 }}{2} = \frac{{3 + 4\sqrt 3 }}{{10}}\\\tan \left( {a + \frac{\pi }{4}} \right) = \frac{{\tan a + \tan \frac{\pi }{4}}}{{1 - \tan a.tan\frac{\pi }{4}}} = \frac{{\frac{4}{3} + 1}}{{1 - \frac{4}{3}}} =  - 7\end{array}\)

\(cos\left(2\pi+\frac{\pi}{16}\right).sin\frac{5\pi}{16}.cos\frac{5\pi}{16}.cos\left(\frac{\pi}{2}-\frac{\pi}{16}\right)\)

\(=\frac{1}{4}.2cos\frac{\pi}{16}.sin\frac{\pi}{16}.2sin\frac{5\pi}{16}.cos\frac{5\pi}{16}\)

\(=\frac{1}{4}sin\frac{2\pi}{16}.sin\frac{10\pi}{16}=\frac{1}{4}sin\frac{\pi}{8}.sin\frac{5\pi}{8}\)

\(=\frac{1}{4}sin\frac{\pi}{8}.sin\left(\frac{\pi}{2}+\frac{\pi}{8}\right)\)

\(=\frac{1}{4}sin\frac{\pi}{8}.cos\frac{\pi}{8}=\frac{1}{8}sin\frac{2\pi}{8}\)

\(=\frac{1}{8}sin\frac{\pi}{4}=\frac{\sqrt{2}}{16}\)

Đề sai hoặc bạn gõ thiếu số 1 ở dưới mẫu

\(A=cos\left(\pi+\frac{\pi}{2}-a\right)-sin\left(\pi+\frac{\pi}{2}-a\right)+cos\left(a+\frac{\pi}{2}-4\pi\right)-sin\left(a+\frac{\pi}{2}-4\pi\right)\)

\(=-cos\left(\frac{\pi}{2}-a\right)+sin\left(\frac{\pi}{2}-a\right)+cos\left(a+\frac{\pi}{2}\right)-sin\left(a+\frac{\pi}{2}\right)\)

\(=-sina+cosa-sina-cosa=-2sina\)

\(sina.sin\left(\frac{\pi}{3}-a\right)sin\left(\frac{\pi}{3}+a\right)\)

\(=-\frac{1}{2}sina\left[cos\frac{2\pi}{3}-cos2a\right]=-\frac{1}{2}sina\left(-\frac{1}{2}-cos2a\right)\)

\(=\frac{1}{4}sina+\frac{1}{2}sina.cos2a=\frac{1}{4}sina+\frac{1}{4}sin3a-\frac{1}{4}sina\)

\(=\frac{1}{4}sin3a\)

\(sin\frac{\pi}{9}sin\frac{2\pi}{9}sin\frac{4\pi}{9}=sin\frac{\pi}{9}sin\left(\frac{\pi}{3}-\frac{\pi}{9}\right)sin\left(\frac{\pi}{3}+\frac{\pi}{9}\right)=\frac{1}{4}sin\frac{\pi}{3}=\frac{\sqrt{3}}{8}\)

\(cosa.cos\left(\frac{\pi}{3}-a\right)cos\left(\frac{\pi}{3}+a\right)=\frac{1}{2}cosa\left(cos\frac{2\pi}{3}+cos2a\right)\)

\(=\frac{1}{2}cosa\left(cos2a-\frac{1}{2}\right)=\frac{1}{2}cosa.cos2a-\frac{1}{4}cosa\)

\(=\frac{1}{4}cos3a+\frac{1}{4}cosa-\frac{1}{4}cosa=\frac{1}{4}cos3a\)

\(cos\frac{\pi}{18}cos\frac{5\pi}{18}cos\frac{7\pi}{18}=cos\frac{\pi}{18}.cos\left(\frac{\pi}{3}-\frac{\pi}{18}\right).cos\left(\frac{\pi}{3}+\frac{\pi}{18}\right)=\frac{1}{4}cos\frac{\pi}{6}=\frac{\sqrt{3}}{8}\)

\(cosx+cos\left(x+\frac{\pi}{5}\right)+cos\left(x+\frac{9\pi}{5}\right)+cos\left(x+\frac{2\pi}{5}\right)+cos\left(x+\frac{8\pi}{5}\right)+...+cos\left(x+\frac{5\pi}{5}\right)\)

\(=cosx-2cosx.cos\frac{4\pi}{5}-2cosx.cos\frac{3\pi}{5}-2cosx.cos\frac{2\pi}{5}-2cosx.cos\frac{\pi}{5}-cosx\)

\(=-2cosx\left(cos\frac{\pi}{5}+cos\frac{4\pi}{5}+cos\frac{2\pi}{5}+cos\frac{3\pi}{5}\right)\)

\(=-2cosx\left(2cos\frac{\pi}{2}.cos\frac{3\pi}{10}+2cos\frac{\pi}{2}cos\frac{\pi}{10}\right)\)

\(=0\) (do \(cos\frac{\pi}{2}=0\))

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