Let f(x) = sin(πcos x) and g(x) = cos(2πsin x) be two functions defined for x > 0. Define the following sets whose elements are written in increasing order: X = {x: f(x) = 0}, Y = {x: f'(x) = 0}, Z = {x: g(x) = 0}, W = {x: g'(x) = 0}. List I contains sets X, Y, Z, and W. List II contains some information regarding these sets. Which of the following is the only correct combination?

Correct option is D. IV-(P),(R),(S)
f(x) = 0 ⇒ sin(πcos x) = 0 ⇒ πcos x = nπ ⇒ cos x = n = -1, 0, 1 ⇒ X = {nπ, (2n+1)π/2} = {nπ/2, n ∈ I}
f'(x) = 0 ⇒ cos(πcos x)(-πsin x) = 0
πcos x = (2n+)π/2 or x = nπ ⇒ cos x = n+1/2 or x = nπ ⇒ cos x = ±1/2 or x = nπ
⇒ Y = {…-2π/3, -π/3, 0, π/3, 2π/3, π, 4π/3,…} which is an arithmetic progression
g(x) = 0 ⇒ cos(2πsin x) = 0 ⇒ 2πsin x = (2n+1)π/2 ⇒ sin x = (2n+1)/4 = ±1/4, ±3/4
⇒ sin x = (2n+1)/4 = ±1/4, ±3/4 ⇒ z = {nπ ± sin⁻¹(1/4), nπ ± sin⁻¹(3/4), n ∈ I}
g'(x) = 0 ⇒ -sin(2πsin x)(2πcos x) = 0 ⇒ 2πsin x = nπ or x = (2n+1)π/2 ⇒ sin x = n/2 = 0, ±1/2, ±1 or x = (2n+1)π/2
W = {nπ, (2n+1)π/2, nπ ± π/6, n ∈ I}