Rút gọn biểu thức \(P = \frac{{\sin \left( {2x} \right)\cos \left( x \right)}}{{\left[ {1 + \cos \left( {2x} \right)} \right]\left[ {1 + \cos \left( x \right)} \right]}}\)
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A.
\(P = \tan x\)
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B.
\(P = - \tan \frac{x}{2}\)
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C.
\(P = \cot \frac{x}{2}\)
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D.
\(P = \tan \frac{x}{2}\)
Sử dụng các công thức:
\(\sin \left( {2a} \right) = 2\sin \left( a \right)\cos \left( a \right)\);
\(\tan x = \frac{{\sin x}}{{\cos x}}\);
\(1 + \cos \left( {2a} \right) = 2{\cos ^2}\left( a \right)\)
\(\begin{array}{l}P = \frac{{\sin \left( {2x} \right)\cos \left( x \right)}}{{\left[ {1 + \cos \left( {2x} \right)} \right]\left[ {1 + \cos \left( x \right)} \right]}} = \frac{{2\sin \left( x \right){{\cos }^2}\left( x \right)}}{{2{{\cos }^2}\left( x \right)\left[ {1 + \cos \left( x \right)} \right]}} = \frac{{\sin \left( x \right)}}{{2{{\cos }^2}\left( {\frac{x}{2}} \right)}}\\ = \frac{{2\sin \left( {\frac{x}{2}} \right)\cos \left( {\frac{x}{2}} \right)}}{{2{{\cos }^2}\left( {\frac{x}{2}} \right)}} = \frac{{2\sin \left( {\frac{x}{2}} \right)}}{{2\cos \left( {\frac{x}{2}} \right)}} = \tan \left( {\frac{x}{2}} \right)\end{array}\)
Đáp án : D
