The equation  $\sin \left(\cos x\right)=\cos \left(\sin x\right)$ has

$\sin \left(\cos x\right)=\cos \left(\sin x\right)$

$\Rightarrow \cos (\frac{\pi }{2}-\cos x)=\cos \left(\sin x\right)$

$\Rightarrow \frac{\pi }{2}-\cos x=2n\pi +\sin x,n\in Z$

$\Rightarrow \cos x+\sin x=\frac{\pi }{2}-2n\pi$

$\Rightarrow \cos x-\sin x=\frac{\pi }{2}-2n\pi$

This gives  $\sin x=0$ and  $\cos x=\frac{\pi }{2}-2n\pi$ which is impossible,

because  $\sin x=0$ $\Rightarrow \cos x=\pm 1$.