Rút gọn biểu thức \(\cos {54^0}\cos {4^0} - \cos {36^0}\cos {86^0}\)ta được:
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A.
\( - \cos \left( {{{58}^0}} \right)\)
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B.
\(\sin \left( {{{58}^0}} \right)\)
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C.
\(\cos \left( {{{58}^0}} \right)\)
-
D.
\( - \sin \left( {{{58}^0}} \right)\)
Sử dụng công thức:
\(\cos \left( a \right)\cos \left( b \right) = \frac{1}{2}\left[ {\cos \left( {a + b} \right) + \cos \left( {a - b} \right)} \right]\).
\(\cos \left( a \right) - \cos \left( b \right) = - 2\sin \left( {\frac{{a + b}}{2}} \right)\sin \left( {\frac{{a - b}}{2}} \right)\).
\(\sin \left( a \right) = \cos \left( {90 - a} \right)\).
\(\cos {54^0}\cos {4^0} - \cos {36^0}\cos {86^0} = \frac{1}{2}\left[ {\cos {{58}^0} + \cos {{50}^0}} \right] - \frac{1}{2}\left[ {\cos {{122}^0} + \cos {{50}^0}} \right]\)
\( = \frac{1}{2}\left( {\cos {{58}^0} + \cos {{50}^0} - \cos {{122}^0} - \cos {{50}^0}} \right) = \frac{1}{2}\left( {\cos {{58}^0} - \cos {{122}^0}} \right)\)
\( = \frac{1}{2}.\left( { - 2} \right)\sin \left( {{{90}^0}} \right)\sin \left( { - {{32}^0}} \right) = \sin \left( {{{32}^0}} \right) = \sin \left( {90 - {{58}^0}} \right) = \cos \left( {{{58}^0}} \right)\)
Đáp án : C
