Nếu t a n ( α ) và t a n ( β ) là nghiệm của phương trình x 2 − p x + q = 0 , ( q ≠ 1 ) thì giá trị của biểu thức Q = c o s 2 ( α + β ) + p s i n ( α + β ) c o s ( α + β ) + q s
Ta có \[{\rm{tan}}\left( {\rm{\alpha }} \right)\]và \[{\rm{tan}}\left( {\rm{\beta }} \right)\]là nghiệm của phương trình \[{{\rm{x}}^{\rm{2}}} - {\rm{px + q = 0, (q}} \ne 1)\]
Theo định lí Vi-ét ta có:
\(\left\{ {\begin{array}{*{20}{c}}{{\rm{tan(\alpha ) + tan(\beta ) = p}}}\\{{\rm{tan(\alpha )}}{\rm{.tan(\beta ) = q}}}\end{array}} \right. \Rightarrow tan(\alpha + \beta )\,\,{\rm{ = }}\frac{{{\rm{tan(\alpha ) + tan(\beta )}}}}{{{\rm{1}} - {\rm{tan(\alpha )}}{\rm{.tan(\beta )}}}}{\rm{ = }}\frac{p}{{1 - q}}\)
\[ \Rightarrow {\cos ^2}\left( {\alpha + \beta } \right){\rm{ = }}\frac{1}{{1 + ta{n^2}(\alpha + \beta )}}{\rm{ = }}\frac{1}{{1 + \frac{{{p^2}}}{{{{\left( {1 - q} \right)}^2}}}}}{\rm{ = }}\frac{{{{\left( {1 - q} \right)}^2}}}{{{{\left( {1 - q} \right)}^2} + {p^2}}}\]
\[q \ne 1 \Rightarrow \frac{{sin(\alpha )sin(\beta )}}{{cos(\alpha )cos(\beta )}} \ne 1 \Rightarrow sin(\alpha )sin(\beta ) \ne cos(\alpha )cos(\beta )\]
\[ \Rightarrow {\rm{cos(\alpha + \beta ) = cos(\alpha )cos(\beta )}} - {\rm{sin(\alpha )sin(\beta )}} \ne 0\]
\( \Rightarrow Q\,\,{\rm{ = }}co{s^2}(\alpha + \beta )\left[ {1 + p.\frac{{sin(\alpha + \beta )}}{{cos(\alpha + \beta )}} + q.\frac{{si{n^2}(\alpha + \beta )}}{{co{s^2}(\alpha + \beta )}}} \right]\)
\[{\rm{ = co}}{{\rm{s}}^{\rm{2}}}\left( {{\rm{\alpha + \beta }}} \right)\left[ {{\rm{1 + p}}{\rm{.tan}}\left( {{\rm{\alpha + \beta }}} \right){\rm{ + q}}{\rm{.ta}}{{\rm{n}}^{\rm{2}}}\left( {{\rm{\alpha + \beta }}} \right)} \right]\]
\({\rm{ = }}\frac{{{{\left( {1 - q} \right)}^2}}}{{{{\left( {1 - q} \right)}^2} + {p^2}}}\left[ {1 + \frac{{{p^2}}}{{1 - q}} + \frac{{{p^2}q}}{{{{\left( {1 - q} \right)}^2}}}} \right]{\rm{ = }}\frac{{{{\left( {1 - q} \right)}^2}}}{{{{\left( {1 - q} \right)}^2} + {p^2}}}.\frac{{{{\left( {1 - q} \right)}^2} + {p^2}\left( {1 - q} \right) + {p^2}q}}{{{{\left( {1 - q} \right)}^2}}}\)
\({\rm{ = 1}}\)
Đáp án cần chọn là: D