which pair of functions is identical?

 Here,
$\text{ (a) }{\sin }^{-1}(\sin x)\text{ is defined for }x\in \left[-\frac{\pi }{2},\frac{\pi }{2}\right],\text{ while }\sin \left({\sin }^{-1}x\right)\text{ is defined only for }x\in [-1,1]$

$\text{ (b) }{\log }_e{e}^x\text{, is defined for all }x\text{, while }{e}^{{\log }_{e}x}\text{ is defined for }x>0\text{. }$

$\text{ (c) }{\log }_e{x}^2\text{ is defined for all }x\in R-\left\{0\right\}\text{, while }2{\log }_{e}x\text{ is definedforx>0. }$

Thus, none is identical.