The domain of $f(x)=\sin \left(\log \left(\sqrt{\frac{4-x^2}{1-x}}\right)\right)$ is

$(-2,1) \cup(2, \infty)$

Given $f(x)=\sin \left(\log \sqrt{\frac{4-x^2}{1-x}}\right)$

domain of sin x is R. But domain of log x is x > 0. Hence domain of given function is values of x such that $\sqrt{\frac{4-x^2}{1-x}}>0 \Rightarrow \frac{4-x^2}{1-x}>0$

$\Rightarrow \frac{(x-2)(x+2)}{x-1}>0$

$\Rightarrow x \in(-2,1) \cup(2, \infty)$

Hence (1) is the correct answer.