Rút gọn biểu thức A = c o s 2 ( α ) + c o s 2 ( α + β ) − 2 c o s ( α ) c o s ( β ) c o s ( α + β ) ta được kết quả

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Rút gọn biểu thức\[{\rm{A = co}}{{\rm{s}}^{\rm{2}}}\left( {\rm{\alpha }} \right){\rm{ + co}}{{\rm{s}}^{\rm{2}}}\left( {{\rm{\alpha + \beta }}} \right) - {\rm{2cos}}\left( {\rm{\alpha }} \right){\rm{cos}}\left( {\rm{\beta }} \right){\rm{cos}}\left( {{\rm{\alpha + \beta }}} \right)\]ta được kết quả

\[{\rm{co}}{{\rm{s}}^{\rm{2}}}\left( {\rm{\alpha }} \right)\]

\[{\rm{co}}{{\rm{s}}^{\rm{2}}}\left( {\rm{\beta }} \right)\]

\[{\rm{si}}{{\rm{n}}^{\rm{2}}}\left( {\rm{\alpha }} \right)\]

\[{\rm{si}}{{\rm{n}}^{\rm{2}}}\left( {\rm{\beta }} \right)\]

Giải thích

\[{\rm{A = co}}{{\rm{s}}^{\rm{2}}}\left( {\rm{\alpha }} \right){\rm{ + co}}{{\rm{s}}^{\rm{2}}}\left( {{\rm{\alpha + \beta }}} \right) - {\rm{2cos}}\left( {\rm{\alpha }} \right){\rm{cos}}\left( {\rm{\beta }} \right){\rm{cos}}\left( {{\rm{\alpha + \beta }}} \right)\]

\[{\rm{ = co}}{{\rm{s}}^{\rm{2}}}\left( {\rm{\alpha }} \right){\rm{ + co}}{{\rm{s}}^{\rm{2}}}\left( {{\rm{\alpha + \beta }}} \right) - {\rm{2}}{\rm{.}}\frac{1}{2}\left[ {{\rm{cos}}\left( {{\rm{\alpha + \beta }}} \right) + {\rm{cos}}\left( {{\rm{\alpha }} - {\rm{\beta }}} \right)} \right]{\rm{cos}}\left( {{\rm{\alpha + \beta }}} \right)\]

\[{\rm{ = co}}{{\rm{s}}^{\rm{2}}}\left( {\rm{\alpha }} \right){\rm{ + co}}{{\rm{s}}^{\rm{2}}}\left( {{\rm{\alpha + \beta }}} \right) - {\rm{co}}{{\rm{s}}^{\rm{2}}}\left( {{\rm{\alpha + \beta }}} \right) - {\rm{cos}}\left( {{\rm{\alpha }} - {\rm{\beta }}} \right){\rm{cos}}\left( {{\rm{\alpha + \beta }}} \right)\]

\[ = \frac{{1 - {\rm{cos}}\left( {2{\rm{\alpha }}} \right)}}{2} - \frac{1}{2}\left[ {{\rm{cos}}\left( {2{\rm{\alpha }}} \right) + {\rm{cos}}\left( {2{\rm{\beta }}} \right)} \right]\]

\[ = \frac{1}{2} - \frac{{{\rm{cos}}\left( {2{\rm{\beta }}} \right)}}{2} = {\sin ^2}\left( {\rm{\beta }} \right)\]

Đáp án cần chọn là: D