a) Đúng. Ta có \(\tan \alpha = 3\) nên \(\cot \alpha = \frac{1}{{\tan \alpha }} = \frac{1}{3}\).

b) Đúng. Ta có \(\frac{1}{{{{\cos }^2}\alpha }} = 1 + {\tan ^2}\alpha = 1 + {3^2} = 10\) \[ \Rightarrow {\cos ^2}\alpha = \frac{1}{{10}}\] \( \Leftrightarrow \left[ \begin{array}{l}{\rm{cos}}\alpha = \frac{1}{{\sqrt {10} }}\\{\rm{cos}}\alpha = - \frac{1}{{\sqrt {10} }}\end{array} \right.\).

Vì \({\rm{0}}^\circ < \alpha < 90^\circ \) nên \(\cos \alpha > 0\)\( \Rightarrow {\rm{cos}}\alpha = \frac{1}{{\sqrt {10} }} = \frac{{\sqrt {10} }}{{10}}\).

c) Sai. Vì \[{\sin ^2}\alpha + {\cos ^2}\alpha = 1\]\[ \Rightarrow {\sin ^2}\alpha = {\rm{1}} - {\cos ^2}\alpha = 1 - \frac{1}{{10}} = \frac{9}{{10}}\];

\(\cot \left( {90^\circ - \alpha } \right) = \tan \alpha = 3\).

Suy ra \(5{\sin ^2}\alpha - 3{\cos ^2}\alpha + \cot \left( {90^\circ - \alpha } \right) = 5 \cdot \frac{9}{{10}} - 3 \cdot \frac{1}{{10}} + 3 = \frac{{36}}{5}\).

d) Đúng. Vì \({\rm{tan}}\alpha = 3\) nên \(\cos \alpha \ne 0\).

Chia tử và mẫu của \(E\) cho \({\cos ^2}\alpha \ne 0\), ta được:

\(E = \frac{{\frac{{{{\sin }^2}\alpha }}{{{{\cos }^2}\alpha }} - \frac{{5{{\cos }^2}\alpha }}{{{{\cos }^2}\alpha }}}}{{\frac{{2{{\sin }^2}\alpha }}{{{{\cos }^2}\alpha }} + \frac{{3\sin \alpha \cos \alpha }}{{{{\cos }^2}\alpha }} + \frac{{{{\cos }^2}\alpha }}{{{{\cos }^2}\alpha }}}} = \frac{{{\rm{ta}}{{\rm{n}}^2}\alpha - 5}}{{2{\rm{ta}}{{\rm{n}}^2}\alpha + 3{\rm{tan}}\alpha + 1}}\,\)

\(E = \frac{{9 - 5}}{{18 + 9 + 1}} = \frac{4}{{28}} = \frac{1}{7} = \frac{a}{b} \Rightarrow a = 1,b = 7 \Rightarrow a + b = 8\).

Độ dài cạnh c là: