Đáp án:
$\begin{array}{l}
a){\cos ^2}\alpha + {\sin ^2}\alpha = 1\\
\Rightarrow {\cos ^2}\alpha = 1 - {\left( {\dfrac{2}{3}} \right)^2} = \dfrac{5}{9}\\
\Rightarrow \left\{ \begin{array}{l}
\cos \alpha = \dfrac{{\sqrt 5 }}{3}\\
\tan \alpha = \dfrac{{\sin \alpha }}{{\cos \alpha }} = \dfrac{2}{{\sqrt 5 }} = \dfrac{{2\sqrt 5 }}{5}\\
\cot \alpha = \dfrac{{\sqrt 5 }}{2}
\end{array} \right.\\
b){\cos ^2}\alpha + {\sin ^2}\alpha = 1\\
\Rightarrow {\sin ^2}\alpha = 1 - {\left( {\dfrac{4}{5}} \right)^2} = \dfrac{9}{{25}}\\
\Rightarrow \sin \alpha = \dfrac{3}{5}\\
\Rightarrow \left\{ \begin{array}{l}
\tan \alpha = \dfrac{{\sin \alpha }}{{\cos \alpha }} = \dfrac{3}{4}\\
\cot \alpha = \dfrac{4}{3}
\end{array} \right.\\
c){\tan ^2}\alpha + 1 = \dfrac{1}{{{{\cos }^2}\alpha }}\\
\Rightarrow \dfrac{1}{{{{\cos }^2}\alpha }} = {\left( {\dfrac{5}{{12}}} \right)^2} + 1 = \dfrac{{169}}{{144}}\\
\Rightarrow {\cos ^2}\alpha = \dfrac{{144}}{{169}}\\
\Rightarrow \cos \alpha = \dfrac{{12}}{{13}}\\
\Rightarrow \sin \alpha = \dfrac{5}{{13}}\\
\Rightarrow \cot \alpha = \dfrac{{12}}{5}\\
d)\cot \alpha = \dfrac{4}{3}\\
\Rightarrow \tan \alpha = \dfrac{3}{4}\\
\Rightarrow \dfrac{1}{{{{\cos }^2}\alpha }} = \dfrac{{25}}{{16}}\\
\Rightarrow \cos \alpha = \dfrac{4}{5}\\
\Rightarrow \sin \alpha = \dfrac{3}{5}
\end{array}$
