Đáp án:
\[\tan 2a = \dfrac{{24}}{7}\]
Giải thích các bước giải:
\(\begin{array}{l}
90^\circ < a < 270^\circ \Rightarrow \cos a < 0\\
\sin a - \cos a = \dfrac{1}{5} \Rightarrow \sin a = \cos a + \dfrac{1}{5}\\
{\sin ^2}a + {\cos ^2}a = 1\\
\Leftrightarrow {\left( {\cos a + \dfrac{1}{5}} \right)^2} + {\cos ^2}a = 1\\
\Leftrightarrow {\cos ^2}a + \dfrac{2}{5}\cos a + \dfrac{1}{{25}} + {\cos ^2}a = 1\\
\Leftrightarrow 2{\cos ^2}a + \dfrac{2}{5}\cos a - \dfrac{{24}}{{25}} = 0\\
\Leftrightarrow \left[ \begin{array}{l}
\cos a = \dfrac{3}{5}\\
\cos a = - \dfrac{4}{5}
\end{array} \right.\\
\cos a < 0 \Rightarrow \cos a = - \dfrac{4}{5} \Rightarrow \sin a = - \dfrac{3}{5}\\
\sin 2a = 2\sin a.\cos a = 2.\left( { - \dfrac{3}{5}} \right).\left( { - \dfrac{4}{5}} \right) = \dfrac{{24}}{{25}}\\
\cos 2a = 2{\cos ^2}a - 1 = 2.{\left( { - \dfrac{4}{5}} \right)^2} - 1 = \dfrac{7}{{25}}\\
\tan 2a = \dfrac{{\sin 2a}}{{\cos 2a}} = \dfrac{{24}}{7}
\end{array}\)
