Đáp án:
a. \(\dfrac{2}{3}{\tan ^6}a\)
Giải thích các bước giải:
\(\begin{array}{l}
b.Do:a \in \left( {\dfrac{{3\pi }}{2};2\pi } \right)\\
\to \cos a > 0\\
Do:\sin a = - \dfrac{4}{{11}}\\
{\sin ^2}a + {\cos ^2}a = 1\\
\to \dfrac{{16}}{{121}} + {\cos ^2}a = 1\\
\to {\cos ^2}a = \dfrac{{105}}{{121}}\\
\to \cos a = \dfrac{{\sqrt {105} }}{{11}}\\
a.A = \dfrac{{2.\left( {\dfrac{{{{\sin }^2}a}}{{{{\cos }^2}a}} - {{\sin }^2}a} \right)}}{{3\left( {\dfrac{{{{\cos }^2}a}}{{{{\sin }^2}a}} - {{\cos }^2}a} \right)}}\\
= 2.\left( {\dfrac{{{{\sin }^2}a - {{\sin }^2}a.{{\cos }^2}a}}{{{{\cos }^2}a}}} \right):3\left( {\dfrac{{{{\cos }^2}a - {{\sin }^2}a.{{\cos }^2}a}}{{{{\sin }^2}a}}} \right)\\
= 2.\left( {\dfrac{{{{\sin }^2}a - {{\sin }^2}a.{{\cos }^2}a}}{{{{\cos }^2}a}}} \right).\dfrac{{{{\sin }^2}a}}{{3\left( {{{\cos }^2}a - {{\sin }^2}a.{{\cos }^2}a} \right)}}\\
= \dfrac{{2\left( {{{\sin }^4}a - {{\sin }^4}a.{{\cos }^2}a} \right)}}{{3\left( {{{\cos }^4}a - {{\sin }^2}a.{{\cos }^4}a} \right)}}\\
= \dfrac{{2{{\sin }^4}a\left( {1 - {{\cos }^2}a} \right)}}{{3{{\cos }^4}a\left( {1 - {{\sin }^2}a} \right)}}\\
= \dfrac{{2{{\sin }^4}a.{{\sin }^2}a}}{{3{{\cos }^4}a{{\cos }^2}a}} = \dfrac{{2{{\sin }^6}a}}{{3{{\cos }^6}a}}\\
= \dfrac{2}{3}{\tan ^6}a
\end{array}\)
