\[\begin{array}{l}
y = 4{\cos ^4}x + 4{\sin ^2}x - 5\\
= 4{\cos ^4}x - 4{\cos ^2}x + 4 - 5\\
= 4{\cos ^4}x - 4{\cos ^2}x + 1 - 2\\
= {\left( {2{{\cos }^2}x - 1} \right)^2} - 1 \ge - 1\\
\Rightarrow Min\,y = - 1\,\,\,khi\,\,\,{\cos ^2}x = \frac{1}{2}.
\end{array}\]