Let f(x)= $\sin \left(\pi \cos x\right)$ and g(x)=$\cos \left(1\pi \sin x\right)$ e two function defined for $x>0$ define the following sets whose elements are written in increasing order.
$X=\left\{x:f\left(x\right)=0\right\},Y=\left\{x:f^{\prime }\left(x\right)=0\right\}$
$Z=\left\{x:g\left(x\right)=0\right\},W=\left\{x:g^{\prime }\left(x\right)=0\right\}$
$\begin{array}{cc}\text{List I}& \text{List II}\\ \left(I\right)X& \left(P\right)\supseteq \left\{\frac{\pi }{2},\frac{3\pi }{2},4\pi ,7\pi \right\}\\ \left(II\right)Y& \left(Q\right)\text{ an arithmetic progression}\\ \left(III\right)Z& \left(R\right)\text{ not an arithmetic progression}\\ \left(IV\right)\text{ W}& \left(S\right)\supseteq \left\{\frac{\pi }{6},\frac{7\pi }{6},\frac{13\pi }{6}\right\}\\ & \left(T\right)\supseteq \left\{\frac{\pi }{3},\frac{2\pi }{3},\pi \right\}\\ & \left(U\right)\supseteq \left\{\frac{\pi }{6},\frac{3\pi }{4}\right\}\end{array}$
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