Trong các khẳng định sau, khẳng định nào sai?

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Trong các khẳng định sau, khẳng định nào sai?

\(\frac{{\tan 30^\circ + \tan 40^\circ + \tan 50^\circ + \tan 60^\circ }}{{cos20^\circ }} = \frac{4}{{\sqrt 3 }}\);

\(cos\frac{\pi }{5} - cos\frac{{2\pi }}{5} = \frac{1}{2}\);

\(cos\frac{\pi }{7} - cos\frac{{2\pi }}{7} + cos\frac{{3\pi }}{7} = \frac{1}{2}\);

\(cos\frac{{2\pi }}{5} + cos\frac{{4\pi }}{5} + cos\frac{{6\pi }}{5} + cos\frac{{8\pi }}{5} = - 1\).

Giải thích

Hướng dẫn giải:

Đáp án đúng là: A

Xét đáp án A

\(\frac{{\tan 30^\circ + \tan 40^\circ + \tan 50^\circ + \tan 60^\circ }}{{cos20^\circ }}\)

\( = \frac{{\frac{{\sin \left( {30^\circ + 40^\circ } \right)}}{{cos30^\circ cos40^\circ }} + \frac{{\sin \left( {50^\circ + 60^\circ } \right)}}{{cos50^\circ cos60^\circ }}}}{{cos20^\circ }}\)

\( = \frac{{\frac{{\sin 70^\circ }}{{cos30^\circ cos40^\circ }} + \frac{{\sin 110^\circ }}{{cos50^\circ cos60^\circ }}}}{{cos20^\circ }}\)

\( = \frac{{\sin 70^\circ }}{{cos20^\circ cos30^\circ cos40^\circ }} + \frac{{\sin 110^\circ }}{{cos20^\circ cos50^\circ cos60^\circ }}\)

\( = \frac{{\sin 70^\circ }}{{\sin 70^\circ cos30^\circ cos40^\circ }} + \frac{{cos20^\circ }}{{cos20^\circ cos50^\circ cos60^\circ }}\)

\( = \frac{1}{{cos30^\circ cos40^\circ }} + \frac{1}{{cos50^\circ cos60^\circ }}\)

\( = \frac{2}{{\sqrt 3 cos40^\circ }} + \frac{2}{{cos50^\circ }}\)

\( = 2 \cdot \frac{{cos50^\circ + \sqrt 3 cos40^\circ }}{{\sqrt 3 cos40^\circ cos50^\circ }}\)

\( = 2 \cdot \frac{{\sin 40^\circ + \sqrt 3 cos40^\circ }}{{\sqrt 3 cos40^\circ cos50^\circ }}\)

\( = 4 \cdot \frac{{\frac{1}{2} \cdot \sin 40^\circ + \frac{{\sqrt 3 }}{2} \cdot cos40^\circ }}{{\frac{{\sqrt 3 }}{2}\left( {cos10^\circ + cos90^\circ } \right)}}\)

\( = 4 \cdot \frac{{\sin 30^\circ \cdot \sin 40^\circ + co{\mathop{\rm s}\nolimits} 30^\circ \cdot cos40^\circ }}{{\frac{{\sqrt 3 }}{2}\left( {cos10^\circ + cos90^\circ } \right)}}\)

\( = \frac{8}{{\sqrt 3 }} \cdot \frac{{cos10^\circ }}{{cos10^\circ }} = \frac{8}{{\sqrt 3 }}\).

Vậy đáp án A sai

Xét đáp án B

\(cos\frac{\pi }{5} - cos\frac{{2\pi }}{5}\)\( = 2\sin \frac{{3\pi }}{{10}}\sin \frac{\pi }{{10}}\)\( = 2\sin \left( {\frac{\pi }{2} - \frac{\pi }{5}} \right)\sin \left( {\frac{\pi }{2} - \frac{{2\pi }}{5}} \right)\)\( = 2cos\frac{\pi }{5}{\rm{cos}}\frac{{2\pi }}{5}\)

\( = \frac{{2cos\frac{\pi }{5}{\rm{cos}}\frac{{2\pi }}{5}\sin \frac{\pi }{5}}}{{\sin \frac{\pi }{5}}}\)\( = \frac{{\sin \frac{{2\pi }}{5}{\rm{cos}}\frac{{2\pi }}{5}}}{{\sin \frac{\pi }{5}}}\)\( = \frac{1}{2} \cdot \frac{{\sin \frac{{4\pi }}{5}}}{{\sin \frac{\pi }{5}}}\)\[ = \frac{1}{2} \cdot \frac{{\sin \left( {\pi - \frac{\pi }{5}} \right)}}{{\sin \frac{\pi }{5}}}\]\[ = \frac{1}{2} \cdot \frac{{\sin \frac{\pi }{5}}}{{\sin \frac{\pi }{5}}} = \frac{1}{2}\].

Vậy đáp án B đúng.

Xét đáp án C

\(cos\frac{\pi }{7} - cos\frac{{2\pi }}{7} + cos\frac{{3\pi }}{7}\)\( = \frac{{\sin \frac{\pi }{7}\left( {cos\frac{\pi }{7} - cos\frac{{2\pi }}{7} + cos\frac{{3\pi }}{7}} \right)}}{{\sin \frac{\pi }{7}}}\)

\( = \frac{{\sin \frac{\pi }{7}cos\frac{\pi }{7} - \sin \frac{\pi }{7}cos\frac{{2\pi }}{7} + \sin \frac{\pi }{7}cos\frac{{3\pi }}{7}}}{{\sin \frac{\pi }{7}}}\)

\( = \frac{1}{2} \cdot \frac{{\sin \frac{{2\pi }}{7} + \sin \frac{\pi }{7} - sin\frac{{3\pi }}{7} - \sin \frac{{2\pi }}{7} + \sin \frac{{4\pi }}{7}}}{{\sin \frac{\pi }{7}}}\)

\( = \frac{1}{2} \cdot \frac{{\sin \frac{\pi }{7} - sin\left( {\pi - \frac{{4\pi }}{7}} \right) + \sin \frac{{4\pi }}{7}}}{{\sin \frac{\pi }{7}}}\)

\( = \frac{1}{2} \cdot \frac{{\sin \frac{\pi }{7} - sin\frac{{4\pi }}{7} + \sin \frac{{4\pi }}{7}}}{{\sin \frac{\pi }{7}}} = \frac{1}{2}\).

Vậy đáp án C đúng

Xét đáp án D

\(cos\frac{{2\pi }}{5} + cos\frac{{4\pi }}{5} + cos\frac{{6\pi }}{5} + cos\frac{{8\pi }}{5}\)

\( = \frac{{2\sin \frac{{2\pi }}{5}cos\frac{{2\pi }}{5} + 2\sin \frac{{2\pi }}{5}cos\frac{{4\pi }}{5} + 2\sin \frac{{2\pi }}{5}cos\frac{{6\pi }}{5} + 2\sin \frac{{2\pi }}{5}cos\frac{{8\pi }}{5}}}{{2\sin \frac{{2\pi }}{5}}}\)

\( = \frac{{\sin \frac{{4\pi }}{5} - \sin \frac{{2\pi }}{5} + \sin \frac{{6\pi }}{5} - \sin \frac{{4\pi }}{5} + \sin \frac{{8\pi }}{5} - \sin \frac{{6\pi }}{5} + \sin 2\pi }}{{2\sin \frac{{2\pi }}{5}}}\)

\( = \frac{{ - \sin \frac{{2\pi }}{5} + \sin \frac{{8\pi }}{5}}}{{2\sin \frac{{2\pi }}{5}}}\)\( = \frac{{2cos\pi \sin \frac{{3\pi }}{5}}}{{2\sin \frac{{2\pi }}{5}}}\)\( = \frac{{ - 2\sin \left( {\pi - \frac{{2\pi }}{5}} \right)}}{{2\sin \frac{{2\pi }}{5}}}\)\( = \frac{{ - 2\sin \frac{{2\pi }}{5}}}{{2\sin \frac{{2\pi }}{5}}} = - 1\).

Vậy đáp án D đúng.