答案
$\sqrt{2}$
解析
解 $\because \frac{\mathrm{d} x}{\mathrm{~d} t}=\cos t, \quad \frac{\mathrm{d} y}{\mathrm{~d} t}=t \cos t$,
$$
\begin{gathered}
\therefore \frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{t \cos t}{\cos t}=t, \\
\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}=\frac{\mathrm{d}}{\mathrm{d} x}\left(\frac{\mathrm{d} y}{\mathrm{~d} x}\right)=\frac{\frac{\mathrm{d}\left(\frac{\mathrm{d} y}{\mathrm{~d} x}\right)}{\mathrm{d} t}}{\frac{\mathrm{d} x}{\mathrm{~d} t}}=\frac{1}{\cos t}, \\
\text { 从而 }\left.\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}\right|_{t=\frac{\pi}{4}}=\frac{1}{\cos \frac{\pi}{4}}=\sqrt{2} .
\end{gathered}
$$