$\begin{array}{l}
y = {\sin ^4}x - 4{\sin ^2}x + 5\\
= {\left( {1 - {{\cos }^2}x} \right)^2} - 4\left( {1 - {{\cos }^2}x} \right) + 5\\
= 1 - 2{\cos ^2}x + {\cos ^4}x - 4 + 4{\cos ^2}x + 5\\
= {\cos ^4}x + 2{\cos ^2}x + 2\\
Ta\,\,co:\,\,\,0 \le {\cos ^2}x \le 1;\,\,\,\,0 \le {\cos ^4}x \le 1\\
\Rightarrow \left\{ \begin{array}{l}
0 \le {\cos ^4}x \le 1\\
0 \le 2{\cos ^2}x \le 2
\end{array} \right. \Rightarrow 0 \le {\cos ^4}x + 2{\cos ^2}x \le 3\\
\Rightarrow 2 \le {\cos ^4}x + 2{\cos ^2}x + 2 \le 5\\
\Rightarrow \left\{ \begin{array}{l}
Miny = 2\,\, \Leftrightarrow {\cos ^2}x = 0 \Leftrightarrow \cos x = 0\\
Max\,\,y = 5 \Leftrightarrow {\cos ^2}x = 1 \Leftrightarrow \sin x = 0
\end{array} \right..
\end{array}$
