Mathematics - Trigonometric Equation Question with Solution | TestHub

$f(x)=x^2,g(x)=\sin x$ $(g\circ f)(x)=\sin x^2$ go(gof) $(x)=\sin \left(\sin x^2\right)$ $(f\circ g\circ g\circ f)(x)={\left(\sin \left(\sin x^2\right)\right)}^2$ Again, $(g\circ f)(x)=\sin x^2$ (gogof) $(x)=\sin \left(\sin x^2\right)$ Given, $($ fogogof $)(x)=($ gogof $)(x)$ $\Rightarrow {\left(\sin \left(\sin x^2\right)\right)}^2=\sin \left(\sin x^2\right)$ $\Rightarrow \sin \left(\sin x^2\right)\left\{\sin \left(\sin x^2\right)-1\right\}=0$ $\Rightarrow \sin \left(\sin x^2\right)=0$ or $\sin \left(\sin x^2\right)=1$ $\Rightarrow \sin x^2=0$ or $\sin x^2=\frac{\pi }{2}$ $\therefore x^2=n\pi$ (i.e. not possible as $-1\le \sin \theta \le 1$ )