Let f(x) = (π cos x) and g(x) = (2π sin x) be two functions defined...

$\left(I\right)$f\left(x\right) = \sin \left(\pi cos x\right)$and$X = \left\{x: f\left(x\right) = 0\right\}$<br />$f\left(x\right) = 0$<br />$⇒ \sin \left(\pi cos x\right) = 0$<br />$⇒ \pi cos x = m\pi, m \in I$<br />$⇒ cos x = m$<br />$cos x = 0, 1, - 1$<br />$⇒ x = \dfrac{n\pi}{2}$<br />$⇒ X = \left\{\dfrac{n\pi}{2}, n \in I\right\} ⇒ X = \left\{\pm \dfrac{\pi}{2}, \pm \pi, \pm \dfrac{3\pi}{2}, \pm 2\pi \ldots \ldots\right\}$<br />\left(I\right) - P, Q$
\left(II\right)g\left(x\right) = \cos \left(2\pi sin x\right)$and$Z = \left\{x: g\left(x\right) = 0\right\}$<br />$\cos \left(2\pi sin x\right) = 0$<br />$⇒ 2\pi sin x = \left(2m + 1\right)\dfrac{\pi}{2}, m \in I$<br />$⇒ sin x = \dfrac{2n + 1}{4}$<br />$⇒ sin x = - \dfrac{1}{4}, \dfrac{1}{4}, \dfrac{- 3}{4}, \dfrac{3}{4}$<br />$⇒ Z = \left\{n\pi \pm \sin^{- 1} \left(\dfrac{1}{4}\right), n\pi \pm \sin^{- 1} \left(\dfrac{3}{4}\right)\right\}, n \in I$<br />\left(II\right) - Q, T$
\left(III\right)f\left(x\right) = \sin \left(\pi cos x\right)$<em>and</em>$Y = \left(x: {f}^{'}\left(x\right) = 0\right)$<br />${f}^{'}\left(x\right) = \cos \left(\pi cos x\right) . \left(- \pi sin x\right) = 0$<br />Now,$\cos \left(\pi cos x\right) = 0 ⇒ \pi cos x = \left(2m + 1\right)\dfrac{\pi}{2}, m \in I$<br />$⇒ cos x = \dfrac{2m + 1}{2}$<br />$⇒ cos x = - \dfrac{1}{2}, \dfrac{1}{2} ⇒ x = n\pi \pm \dfrac{\pi}{3}, n \in I$<br />or$sin x = 0 ⇒ x = n\pi$<br />Hence,$Y = \left\{n\pi, n\pi \pm \dfrac{\pi}{3}\right\}, n \in I$<br />$Y = \left\{\dfrac{\pi}{3}, \dfrac{2\pi}{3}, \pi, \dfrac{4\pi}{3}, \dfrac{5\pi}{3}, 2\pi \ldots\right\}$<br />\left(III\right) \to R$
\left(IV\right)$g\left(x\right) = \cos \left(2\pi sin x\right)$ and $W = \left\{x: {g}^{'}\left(x\right) = 0\right\}$
${g}^{'}\left(x\right) = - \sin \left(2\pi sin x\right)\left(2\pi cos x\right) = 0$
Now, $cos x = 0 ⇒ x = \left(2m + 1\right)\dfrac{\pi}{2}, m \in I$
or $\sin \left(2\pi sin x\right) = 0$
$2\pi sin x = m\pi, m \in I$
$⇒ sin x = \dfrac{m}{2}$
$⇒ sin x = - 1, - \dfrac{1}{2}, 0, \dfrac{1}{2}, 1$
$⇒ x = \dfrac{n\pi}{2}, n\pi \pm \dfrac{\pi}{6}, n \in I$
$W = \left\{\dfrac{n\pi}{2}, n\pi \pm \dfrac{\pi}{6}\right\}$
$W = \left\{\dfrac{\pi}{6}, \dfrac{\pi}{2}, \dfrac{5\pi}{6}, \pi, \dfrac{7\pi}{6}, \dfrac{3\pi}{2} \ldots .\right\}$
\left(IV\right) - P, R, S