Đáp án:
\[E = {\tan ^6}a\]
Giải thích các bước giải:
Ta có:
\(\begin{array}{l}
E = \dfrac{{{{\sin }^2}a - {{\tan }^2}a}}{{{{\cos }^2}a - {{\cot }^2}a}}\\
= \dfrac{{{{\sin }^2}a - \dfrac{{{{\sin }^2}a}}{{{{\cos }^2}a}}}}{{{{\cos }^2}a - \dfrac{{{{\cos }^2}a}}{{{{\sin }^2}a}}}}\\
= \dfrac{{{{\sin }^2}a\left( {1 - \dfrac{1}{{{{\cos }^2}a}}} \right)}}{{{{\cos }^2}a\left( {1 - \dfrac{1}{{{{\sin }^2}a}}} \right)}}\\
= \dfrac{{{{\sin }^2}a.\dfrac{{{{\cos }^2}a - 1}}{{{{\cos }^2}a}}}}{{{{\cos }^2}a.\dfrac{{{{\sin }^2}a - 1}}{{{{\sin }^2}a}}}}\\
= \dfrac{{{{\sin }^2}a.\dfrac{{ - {{\sin }^2}a}}{{{{\cos }^2}a}}}}{{{{\cos }^2}a.\dfrac{{ - {{\cos }^2}a}}{{{{\sin }^2}a}}}}\\
= \dfrac{{{{\sin }^4}a}}{{{{\cos }^2}a}}:\dfrac{{{{\cos }^4}a}}{{{{\sin }^2}a}}\\
= \dfrac{{{{\sin }^6}a}}{{{{\cos }^6}a}} = {\tan ^6}a
\end{array}\)
