Giải các phương trình :            $\begin{array}{l}1)\,\,\,{3^{\left| {\sin \sqrt x } \right|}} = \left| {\cos x} \right|\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(1)\\2)\,\,{\log _2}\left( {co{s^2}xy + \frac{1}{{co{s^2}xy}}} \right) = \frac{1}{{{y^2} – 2y + 2}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(2)\end{array}$

Đề bài:

Giải các phương trình :            $\begin{array}{l}1)\,\,\,{3^{\left| {\sin \sqrt x } \right|}} = \left| {\cos x} \right|\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(1)\\2)\,\,{\log _2}\left( {co{s^2}xy + \frac{1}{{co{s^2}xy}}} \right) = \frac{1}{{{y^2} – 2y + 2}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(2)\end{array}$

Bài giải:

$1)$  Điều kiện    $x \ge 0$
     Do $\left| {\sin \sqrt x  \ge 0} \right|$   nên   ${3^{\left| {\sin \,\sqrt x } \right|}} \ge 1$
           $\left| {\cos x} \right| \le 1$
     Suy ra : $(1)\,\,\,\, \Leftrightarrow \,\,\,\left\{ \begin{array}{l}
\sin \,\sqrt x  = 0\\
\cos x =  \pm 1
\end{array} \right.\,\,\,\,\,\,\,\,\, \Leftrightarrow \,\,\left\{ \begin{array}{l}
\sqrt x  = k\pi \\
x = h\pi
\end{array} \right.\,\,\,\,\,\,\,(k,h \in {\rm Z})$
    $ \Leftrightarrow \left\{ \begin{array}{l}
x = {k^2}{\pi ^2}\\
x = h\pi
\end{array} \right.\,\,\,\,\,\,\,\,\, \Leftrightarrow \left\{ \begin{array}{l}
h = {k^2}\pi \\
x = {k^2}\pi
\end{array} \right.(1′)\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,,\,\,\,\,\
\\h \in {\rm Z}  nên (1′) \Leftrightarrow \,\,\,\left\{ \begin{array}{l}
k = 0\\
x = 0
\end{array} \right.$

    Vậy nghiệm của $(1)$ là $x = 0$
2)      $co{s^2}xy + \frac{1}{{co{s^2}xy}} \ge 2 \Leftrightarrow {\log _2}\left( {co{s^2}xy + \frac{1}{{co{s^2}xy}}} \right) \ge 1$
      Mặt khác   ${y^2} – 2y + 2 = \left( {{y^2} – 2y + 1} \right) + 1 = {\left( {y – 1} \right)^2} \ge 1$
                        $ \Rightarrow \frac{1}{{{y^2} – 2y + 2}} \le 1$
     Do đó $(2)$ $ \Leftrightarrow \left\{ \begin{array}{l}
co{s^2}xy = 1\\
y – 1 = 0
\end{array} \right.\,\,\,\,\,\,\,\,\,\,\, \Leftrightarrow \left\{ \begin{array}{l}
co{s^2}x = 1\\
y = 1
\end{array} \right.\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\left\{ \begin{array}{l}
x = k\pi \\
y = 1
\end{array} \right.\,\,\,\,\,\,\,\,\,\,(k \in {\rm Z})$
     Nghiệm của $(2)$ là $(k\pi ;1)\,\,\,,\,\,\,\,k \in {\rm Z}$