Question

Let $f(x)=x^2$ and $g(x)=\sin x$ for all $x \in R$. Then the set of all $x$ satisfying $(f \circ g \circ g \circ f)(x)=(g \circ g \circ f)(x)$, wheng $(f \circ g)(x)=f(g(x))$, is

$(\mathrm{A}) \pm \sqrt{n \pi}, n \in\{0,1,2, \ldots\}$

(B) $\pm \sqrt{n \pi}, n \in\{1,2, \ldots\}$

(C) $\frac{\pi}{2}+2 n \pi . n \in\{\ldots . .-2,-1,0,1,2, \ldots\}$

(D) $2 n \pi, n \in\{\ldots,-2,-1,0,1,2, \ldots\}$

Answer 1

SOLUTION : $\text { Now } \sin ^2\left(\sin x^2\right)=\sin \left(\sin x^2\right) \Rightarrow \sin \left(\sin x^2\right)=0,1 \\$

$\Rightarrow \quad \sin x^2=n \pi,(4 n+1) \frac{\pi}{2} ; \eta \in I \quad \Rightarrow \quad \sin x^2=0 \\$

$\Rightarrow \quad x^2=n \pi \quad \Rightarrow \quad x= \pm \sqrt{n \pi} ; n \in W \\$

Hence Option A is Correct.