The number of solutions of the equation $\left|\sin \frac{\pi }{2}\left(1-x\right)+\cos \frac{\pi }{2}\left(1-x\right)\right|=\sqrt{\left|{\log }_e{\left|x\right|}^3+1\right|}\forall x>0\text{is}$

The correct option is A 1
Given $\left|\sin \frac{\pi }{2}\left(1-x\right)+\cos \frac{\pi }{2}\left(1-x\right)\right|=\sqrt{\left|{\log }_e{\left|x\right|}^3+1\right|}$
Squaring both sides

$\Rightarrow (\sin \frac{\pi }{2}(1-x){)}^2+(\cos \frac{\pi }{2}(1-x){)}^2+2\sin \frac{\pi }{2}\left(1-x)\cos \frac{\pi }{2}\right(1-x)={\log }_e|x{|}^3+1$
$\Rightarrow \sin (2\times \frac{\pi (1-x)}{2})={\log }_e|x{|}^3$

$\Rightarrow \sin (\pi -\pi x)={\log }_e|x{|}^3$
$\Rightarrow \sin \pi x={\log }_e|x{|}^3$

Now, drawing the graphs for both functions, we get:



For $x>0,$ we have only one solution for the equation.