Step-by-step Solution:

Given that \(\sin (\pi \cos \theta)=\cos (\pi \sin \theta)\) \[\] \(\Rightarrow \sin (\pi \cos \theta)=\sin \left(\frac{\pi}{2} \pm \pi \sin \theta\right)\) \[\] \(\Rightarrow \pi \cos \theta=\frac{\pi}{2} \pm \pi \sin \theta\) \[\] \(\Rightarrow \cos \theta \mp \sin \theta=\frac{1}{2}\) \[\] or \(\cos ^{2} \theta+\sin ^{2} \theta \pm 2 \sin \theta \cos \theta=\frac{1}{4}\) \[\] or \(\pm \sin 2 \theta=-\frac{3}{4} \Rightarrow \sin 2 \theta= \pm \frac{3}{4}\) \[\] Choice (A)\[\]