101 If $\tan (\pi \cos \theta)=\cot (\pi \sin \theta)$, then $\cos \left(\theta-\frac{\pi}{4}\right)=$ $\qquad$
$101 \pm \frac{\sqrt{2}}{4}$. From the given information,
$$
\tan (\pi \cos \theta)=\cot (\pi \sin \theta)=\tan \left(\frac{\pi}{2}-\pi \sin \theta\right),
$$

Therefore,
$$
\pi \cos \theta=k \pi+\frac{\pi}{2}-\pi \sin \theta, k \in \mathbf{Z},
$$

Thus,
$$
\sin \theta+\cos \theta=k+\frac{1}{2},
$$

Hence,
$$
\cos \left(\theta-\frac{\pi}{4}\right)=\frac{k+\frac{1}{2}}{\sqrt{2}}=\frac{2 k+1}{2 \sqrt{2}} .
$$

Also, because
$$
-1 \leqslant \frac{2 k+1}{2 \sqrt{2}} \leqslant 1, k \in \mathbf{Z},
$$

Solving for $k$ yields $k=-1,0$. Therefore, $\cos \left(\theta-\frac{\pi}{4}\right)= \pm \frac{\sqrt{2}}{4}$.