Đáp án:
\[P = \frac{7}{4}\]
Giải thích các bước giải:
Ta có:
\[\begin{array}{l}
\cos x + \sin x = \frac{1}{3} \Leftrightarrow {\left( {\cos x + \sin x} \right)^2} = \frac{1}{9}\\
\Leftrightarrow {\sin ^2}x + {\cos ^2}x + 2\sin x\cos x = \frac{1}{9}\\
\Leftrightarrow \sin x.\cos x = - \frac{4}{9}\\
P = \sqrt {{{\tan }^2}x + {{\cos }^2}x} = \sqrt {\frac{{{{\sin }^2}x}}{{{{\cos }^2}x}} + \frac{{{{\cos }^2}x}}{{{{\sin }^2}x}}} \\
= \sqrt {\frac{{{{\sin }^4}x + {{\cos }^4}x}}{{{{\sin }^2}x.{{\cos }^2}x}}} = \sqrt {\frac{{{{\left( {{{\sin }^2}x + {{\cos }^2}x} \right)}^2} - 2{{\sin }^2}x.{{\cos }^2}x}}{{{{\sin }^2}x.{{\cos }^2}x}}} \\
= \sqrt {\frac{{{1^2} - 2.{{\left( {\frac{{ - 4}}{9}} \right)}^2}}}{{{{\left( { - \frac{4}{9}} \right)}^2}}}} = \frac{7}{4}
\end{array}\]
