2021 AMC 12A Problem 19

How many solutions does the equation \sin \left( \frac{\pi}2 \cos x\right)=\cos \left( \frac{\pi}2 \sin x\right) have in the closed interval [0,\pi]?

\textbf{(A) }0 \qquad \textbf{(B) }1 \qquad \textbf{(C) }2 \qquad \textbf{(D) }3\qquad \textbf{(E) }4

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.