Tìm số nghiệm trên đoạn \[\left[ {{\rm{0; }}\frac{{\rm{\pi }}}{{\rm{2}}}} \right]\]của phương trình\[{\rm{si}}{{\rm{n}}^{\rm{3}}}{\rm{x + sinxcosx = 1}} - {\rm{co}}{{\rm{s}}^{\rm{3}}}{\rm{x}}\]

\[{\rm{si}}{{\rm{n}}^{\rm{3}}}{\rm{x + sinxcosx = 1}} - {\rm{co}}{{\rm{s}}^{\rm{3}}}{\rm{x}} \Leftrightarrow {\rm{si}}{{\rm{n}}^{\rm{3}}}{\rm{x + co}}{{\rm{s}}^{\rm{3}}}{\rm{x = 1}} - {\rm{sinxcosx}}\]

\[ \Leftrightarrow \left( {{\rm{sinx + cosx}}} \right)\left( {{\rm{1}} - {\rm{sinxcosx}}} \right){\rm{ = 1}} - {\rm{sinxcosx}}\]

\( \Leftrightarrow \left[ {\begin{array}{*{20}{c}}{1 - sinxco{\rm{sx = 0}}}\\{{\rm{sinx + cosx = 1}}}\end{array}} \right. \Leftrightarrow \left[ {\begin{array}{*{20}{c}}{{\rm{sin2x = 2(VN)}}}\\{{\rm{sin}}\left( {{\rm{x + }}\frac{{\rm{\pi }}}{{\rm{4}}}} \right){\rm{ = }}\frac{{\rm{1}}}{{\sqrt {\rm{2}} }}}\end{array}} \right. \Leftrightarrow \left[ {\begin{array}{*{20}{c}}{{\rm{x = k2\pi }}}\\{{\rm{x = }}\frac{{\rm{\pi }}}{{\rm{2}}}{\rm{ + k2\pi }}}\end{array}} \right.,k \in \mathbb{Z}\)

Vì \[{\rm{x}} \in \left[ {{\rm{0; }}\frac{{\rm{\pi }}}{{\rm{2}}}} \right] \Rightarrow {\rm{x}} \in \left\{ {{\rm{0; }}\frac{{\rm{\pi }}}{{{\rm{ 2}}}}} \right\}\]Đáp án cần chọn là: C