Range of nested function f(x)

We note that $\forall x \in \mathbb R$ $\implies \sin x \in [-1,1]$ $\implies \sin \left(\dfrac \pi 2 \sin x\right) \in [-1, 1]$ $\implies \sin \left(\dfrac \pi 6 \sin \left(\dfrac \pi 2 \sin x\right)\right) \in [-0.5, 0.5]$ .

If x e R then the range of Sin(X) is [-1,1].

Range of pi/2 * Sin(x) is [-pi/2,pi/2]

Range of Sin(pi/2*Sin(x)) =[-1,1]

Range of pi/6 * Sin(pi/2*Sin(x)) = [-pi/6,pi/6]

Range of Sin (pi/6 * Sin(pi/2*Sin(x)) is hence (-0.5,0.5)