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AMC 12A Spring 2021 (Problem 19)

How many solutions does the equation sin⁡(π2cos⁡x)=cos⁡(π2sin⁡x)\sin\left(\frac{\pi}{2}\cos x\right)=\cos\left(\frac{\pi}{2}\sin x\right) have in the closed interval [0,π][0,\pi]?
(A)  0\text{(A)}\;0(B)  1\text{(B)}\;1(C)  2\text{(C)}\;2(D)  3\text{(D)}\;3(E)  4\text{(E)}\;4