1 The minimum value of the function $y=\sin ^{4} x+\cos ^{4} x+\sec ^{4} x+\csc ^{4} x$ is
18.5.
$$
\begin{aligned}
y & =\sin ^{4} x+\cos ^{4} x+\sec ^{4} x+\csc ^{4} x \\
& \geqslant 2 \sin ^{2} x \cos ^{2} x+2 \sec ^{2} x \csc ^{2} x \\
& =2\left(\frac{1}{4} \sin ^{2} 2 x+\frac{4}{\sin ^{2} 2 x}\right) \\
& =\frac{1}{2}\left(\sin ^{2} 2 x+\frac{16}{\sin ^{2} 2 x}\right)
\end{aligned}
$$

From the fact that $f(x)=x+\frac{16}{x}$ is decreasing on $[0,1]$, we know
$$
y \geqslant \frac{1}{2}\left(1+\frac{16}{1}\right)=8.5 \text {. }
$$

Equality holds when $x=k \pi+\frac{\pi}{4},(k \in \mathbf{Z})$.